diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 1d23887..f8960e4 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -8,15 +8,15 @@ \citation{dvorak-lidicky-cones} \citation{heesch-untersuchungen} \citation{robertson-sanders-seymour-thomas} -\citation{robertson-sanders-seymour-thomas} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent } +\citation{robertson-sanders-seymour-thomas} \newlabel{def:dual}{{1.3}{2}} \newlabel{def:dual-depth}{{1.4}{2}} \newlabel{def:dual-component}{{1.5}{2}} -\newlabel{def:tire-graph}{{1.6}{2}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{3}{}\protected@file@percent } \newlabel{fig:dual-depth}{{1}{3}} +\newlabel{def:tire-graph}{{1.6}{3}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{4}{}\protected@file@percent } \newlabel{fig:tire-example}{{2}{4}} \newlabel{def:medial-tire-graph}{{1.7}{4}} @@ -49,34 +49,6 @@ \newlabel{thm:tread-tree}{{1.23}{14}} \newlabel{rem:tree-multiple-children}{{1.24}{15}} \newlabel{thm:tire-tree-decomposition}{{1.25}{15}} -\newlabel{rem:tree-coloring-factorisation}{{1.26}{17}} -\newlabel{rem:level-cycle-motivation}{{1.27}{17}} -\newlabel{def:level-cycle-three-colour-restriction}{{1.28}{17}} -\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.25\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{18}{}\protected@file@percent } -\newlabel{fig:tire-tree-decomposition}{{6}{18}} -\newlabel{conj:false-universal-level-cycle-three-colour}{{1.29}{18}} -\newlabel{ex:universal-level-cycle-counterexample}{{1.30}{18}} -\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces The $8$-vertex counterexample to the universal-source form. With source $S=\{7\}$, the level cycle $(3,4,5,8)$ lies in $L_2$ and forces all four colours in every proper $4$-vertex-colouring.}}{19}{}\protected@file@percent } -\newlabel{fig:universal-level-cycle-counterexample}{{7}{19}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{An inner-boundary refinement}}{19}{}\protected@file@percent } -\newlabel{def:tire-inner-boundary-three-colour}{{1.31}{19}} -\newlabel{conj:tire-inner-boundary-three-colour}{{1.32}{19}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A counterexample at $n=14$}}{20}{}\protected@file@percent } -\newlabel{ex:inner-boundary-counterexample}{{1.33}{20}} -\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces The $14$-vertex counterexample $G^\star $ to Conjecture\nonbreakingspace 1.32\hbox {} in a planar embedding. The six degree-$3$ vertices split into two triples, $\{3,5,10\}$ each adjacent to a triangle in the core $\{1,2,4,6\}$, and $\{11,13,14\}$ each adjacent to a triangle in the core $\{7,8,9,12\}$; the two cores are joined by the edges $17,28,69$ together with $12$.}}{20}{}\protected@file@percent } -\newlabel{fig:inner-boundary-counterexample}{{8}{20}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{21}{}\protected@file@percent } -\newlabel{conj:level-cycle-three-colour}{{1.34}{21}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{21}{}\protected@file@percent } -\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Exhaustive vertex-source search for the level-cycle three-colour conjecture on all triangulation isomorphism classes with $4 \leq n \leq 13$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.}}{21}{}\protected@file@percent } -\newlabel{tab:level-cycle-three-colour-counts}{{1}{21}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{21}{}\protected@file@percent } -\newlabel{def:seam}{{1.35}{21}} -\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The $5$-connected triangulations at $14 \leq n \leq 24$ generated by \texttt {plantri -c5 -a}. All $9732$ graphs in this slice admit a vertex source witnessing the level-cycle three-colour conjecture.}}{22}{}\protected@file@percent } -\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{22}} -\newlabel{def:partial-tire-tree}{{1.36}{22}} -\newlabel{lem:seam-edge-shared}{{1.37}{22}} -\newlabel{conj:seam-counterexample}{{1.38}{22}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} @@ -93,5 +65,8 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{23}{}\protected@file@percent } -\gdef \@abspage@last{23} +\newlabel{rem:tree-coloring-factorisation}{{1.26}{17}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{17}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.25\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{18}{}\protected@file@percent } +\newlabel{fig:tire-tree-decomposition}{{6}{18}} +\gdef \@abspage@last{18} diff --git a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk index cba488c..a47c09c 100644 --- a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk +++ b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1780943955 "paper.tex" "paper.pdf" "paper" 1780943957 +["pdflatex"] 1780944443 "paper.tex" "paper.pdf" "paper" 1780944444 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -25,14 +25,12 @@ "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti7.tfm" 1136768653 1492 86331993fe614793f5e7e755835c31c5 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 "" - "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm" 1136768653 768 1321e9409b4137d6fb428ac9dc956269 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb" 1248133631 32166 b0c356b15f19587482a9217ce1d8fa67 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb" 1248133631 37912 77d683123f92148345f3fc36a38d9ab1 "" - "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb" 1248133631 37166 8ab3487cbe3ab49ebce74c29ea2418db "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1248133631 35469 70d41d2b9ea31d5d813066df7c99281c "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi9.pfb" 1248133631 36094 798f80770b3b148ceedd006d487db67c "" @@ -49,7 +47,6 @@ "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb" 1248133631 32442 c975af247b6702f7ca0c299af3616b80 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb" 1248133631 35660 fb24af7afbadb71801619f1415838111 "" - "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb" 1248133631 31099 c85edf1dd5b9e826d67c9c7293b6786c "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb" 1248133631 34694 ad62b13721ee8eda1dcc8993c8bd7041 "" "/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b "" @@ -143,13 +140,11 @@ "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" "fig_dual_depth.png" 1779857443 255786 cb48aab5aa40fc161d13a75df0544511 "" - "fig_inner_boundary_counterexample.png" 1780363332 86866 e15e4311e42fec20179ac6bb90683dea "" "fig_medial_tire_example.png" 1780943640 209003 4349824e4f016bde4b938be6b2cb5b2c "" "fig_tire_example.png" 1779857443 104494 8f9ce26b469b4236b8b67829f73a5faa "" "fig_tire_tree_decomposition.png" 1780290287 372371 1b44f5a3e9f637d78ae951b1f2e3a89d "" - "fig_universal_level_cycle_counterexample.png" 1780325973 75145 08f600be4e05c11d702bee45996ca222 "" - "paper.aux" 1780943957 10167 b77d8e1ff2e4a1b03fd4e31e583defb0 "pdflatex" - "paper.tex" 1780943947 86241 c3ce5f87def8730855ab5ccd29b7016c "" + "paper.aux" 1780944444 7237 ad56df69e173b3ef29a32273960fe919 "pdflatex" + "paper.tex" 1780944408 69767 384639d36c3a81504fdd37ae395860bb "" (generated) "paper.aux" "paper.log" diff --git a/papers/coloring_nested_tire_graphs/paper.fls b/papers/coloring_nested_tire_graphs/paper.fls index 0632956..39e0229 100644 --- a/papers/coloring_nested_tire_graphs/paper.fls +++ b/papers/coloring_nested_tire_graphs/paper.fls @@ -477,16 +477,6 @@ INPUT ./fig_tire_tree_decomposition.png INPUT fig_tire_tree_decomposition.png INPUT ./fig_tire_tree_decomposition.png INPUT ./fig_tire_tree_decomposition.png -INPUT ./fig_universal_level_cycle_counterexample.png -INPUT ./fig_universal_level_cycle_counterexample.png -INPUT fig_universal_level_cycle_counterexample.png -INPUT ./fig_universal_level_cycle_counterexample.png -INPUT ./fig_universal_level_cycle_counterexample.png -INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm -INPUT ./fig_inner_boundary_counterexample.png -INPUT fig_inner_boundary_counterexample.png -INPUT ./fig_inner_boundary_counterexample.png -INPUT ./fig_inner_boundary_counterexample.png INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm INPUT paper.aux INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb @@ -495,7 +485,6 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10. INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb -INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi9.pfb @@ -512,6 +501,5 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pf INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb -INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index cec5b01..d150ce5 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 8 JUN 2026 14:39 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 8 JUN 2026 14:47 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -495,25 +495,25 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] - + File: fig_dual_depth.png Graphic file (type png) -Package pdftex.def Info: fig_dual_depth.png used on input line 165. +Package pdftex.def Info: fig_dual_depth.png used on input line 163. (pdftex.def) Requested size: 251.9989pt x 237.67276pt. LaTeX Warning: `h' float specifier changed to `ht'. [2] [3 <./fig_dual_depth.png>] - + File: fig_tire_example.png Graphic file (type png) -Package pdftex.def Info: fig_tire_example.png used on input line 244. +Package pdftex.def Info: fig_tire_example.png used on input line 242. (pdftex.def) Requested size: 280.79956pt x 188.56097pt. - + File: fig_medial_tire_example.png Graphic file (type png) -Package pdftex.def Info: fig_medial_tire_example.png used on input line 278. +Package pdftex.def Info: fig_medial_tire_example.png used on input line 276. (pdftex.def) Requested size: 280.79956pt x 189.42558pt. @@ -525,79 +525,54 @@ LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'. [9] [10] [11] [12] [13] [14] [15] [16] - + File: fig_tire_tree_decomposition.png Graphic file (type png) Package pdftex.def Info: fig_tire_tree_decomposition.png used on input line 13 -34. +32. (pdftex.def) Requested size: 341.9989pt x 196.86678pt. LaTeX Warning: `h' float specifier changed to `ht'. -[17] - -File: fig_universal_level_cycle_counterexample.png Graphic file (type png) - -Package pdftex.def Info: fig_universal_level_cycle_counterexample.png used on -input line 1416. -(pdftex.def) Requested size: 280.79956pt x 156.91663pt. - [18 <./fig_tire_tree_decomposition.png>] [19 <./fig_universal_level_cycle_coun -terexample.png>] - -File: fig_inner_boundary_counterexample.png Graphic file (type png) - -Package pdftex.def Info: fig_inner_boundary_counterexample.png used on input l -ine 1525. -(pdftex.def) Requested size: 280.79956pt x 156.02269pt. - [20 <./fig_inner_boundary_counterexample.png>] [21] -[22] -Overfull \hbox (1.78508pt too wide) in paragraph at lines 1714--1716 -[]\OT1/cmr/m/n/10 Length lower bound (Birkhoff). \OT1/cmr/m/it/10 Ev-ery non-tr -ivial seam $\OML/cmm/m/it/10 C$ \OT1/cmr/m/it/10 of $\OML/cmm/m/it/10 G$ \OT1/c -mr/m/it/10 has $\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/10 (\OML/cmm/m -/it/10 C\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j ^^U - [] - -[23] (./paper.aux) ) +[17] [18 <./fig_tire_tree_decomposition.png>] (./paper.aux) ) Here is how much of TeX's memory you used: - 14117 strings out of 478268 - 281512 string characters out of 5846347 - 565616 words of memory out of 5000000 - 31937 multiletter control sequences out of 15000+600000 - 478386 words of font info for 63 fonts, out of 8000000 for 9000 + 14081 strings out of 478268 + 280188 string characters out of 5846347 + 565505 words of memory out of 5000000 + 31903 multiletter control sequences out of 15000+600000 + 478218 words of font info for 62 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 84i,12n,89p,1168b,801s stack positions out of 10000i,1000n,20000p,200000b,200000s - - -Output written on paper.pdf (23 pages, 1252642 bytes). + +Output written on paper.pdf (18 pages, 1096120 bytes). PDF statistics: - 225 PDF objects out of 1000 (max. 8388607) - 134 compressed objects within 2 object streams + 195 PDF objects out of 1000 (max. 8388607) + 117 compressed objects within 2 object streams 0 named destinations out of 1000 (max. 500000) - 43 words of extra memory for PDF output out of 10000 (max. 10000000) + 33 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 1bc46fc..d36f1f2 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 5f751ee..9f06101 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -28,7 +28,7 @@ \begin{document} -\title{Coloring Nested Tire Graphs} +\title{Nested Tire Decompositions of Plane Triangulations} % author one information \author{Eric Bauerfeld} @@ -46,20 +46,18 @@ \dedicatory{} \begin{abstract} -We establish the foundational structure of nested -level-induced tire decompositions of a plane triangulation $G$. -A \emph{level source} of $G$ induces a BFS layering of $G$ and -endows the inner planar dual $G'$ with a \emph{dual depth} -grading. The basic object of study is the \emph{tire graph} -$T$ --- a plane graph whose outer and inner boundaries bound a -closed planar region, the \emph{tire tread} $R$, triangulated by -the \emph{annular edges} $E_{\mathrm{ann}}$. Our main structural -result, the \emph{tire-component lemma}, exhibits each connected -component of $G'_d$ as a tire graph; the \emph{tire-tread -partition theorem} consequence shows the resulting tire treads -partition the bounded faces of $G$. Coloring questions on -$G$ thereby factor through coloring questions on the -individual treads. +We establish the foundational structure of nested level-induced tire +decompositions of a plane triangulation $G$. A \emph{level source} of +$G$ induces a BFS layering of $G$ and endows the inner planar dual +$G'$ with a \emph{dual depth} grading. The basic object of study is +the \emph{tire graph} $T$ --- a plane graph whose outer and inner +boundaries bound a closed planar region, the \emph{tire tread} $R$, +triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$. We define +medial tire graphs and prove a basic colour-count bound for their +annular medial cycle. Our main structural results are the +\emph{tire-component lemma}, the \emph{tire-tread partition theorem}, +and the rooted \emph{tire-tree decomposition}, which together organize +the bounded faces of $G$ into nested tire treads. \end{abstract} \maketitle @@ -1370,386 +1368,6 @@ This is the structural setup underlying the chain-pigeonhole program for tire treads. \end{remark} -\begin{remark}[Motivation for level-cycle restrictions] -\label{rem:level-cycle-motivation} -The tire-tree decomposition reduces global colouring questions to local -choices on treads together with compatibility along nested boundary -cycles. Without further structure, the number of boundary colour states -can grow quickly as one descends the tree: each seam or level cycle may -in principle carry any proper restriction of a $4$-colouring. The -following restriction is meant to test whether this state space can be -compressed. If level cycles can always be made to omit one colour, then -each such interface behaves like a three-colour boundary object, while -still allowing different cycles to omit different colours. This would -not by itself solve the gluing problem, but it would give a simpler -target class of boundary states for arguments about nested tire trees. -\end{remark} - -\begin{definition}[Level-cycle three-colour restriction] -\label{def:level-cycle-three-colour-restriction} -Let $G$ be a maximal planar graph, let $S \subseteq V(G)$ be a level -source, and let $c \colon V(G) \to \{1,2,3,4\}$ be a proper -$4$-vertex-colouring of $G$. We say that $c$ has the -\emph{level-cycle three-colour restriction} with respect to $S$ if, -for every level $d \geq 0$ and every simple cycle -$C \subseteq G[L_d]$, the colour set used on $C$ has size at most -three: -\[ - |c(V(C))| \leq 3. -\] -Equivalently, every simple cycle contained in a single level omits at -least one of the four colours. The omitted colour may depend on the -cycle; in particular, distinct cycles in the same level, the same tire -tread, or the same inner outerplanar component are not required to -omit the same colour. -\end{definition} - -\begin{conjecture}[False universal-source form] -\label{conj:false-universal-level-cycle-three-colour} -Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be any -level source. Then $G$ admits a proper $4$-vertex-colouring with the -level-cycle three-colour restriction with respect to $S$. -\end{conjecture} - -\begin{figure}[htbp] -\centering -\includegraphics[width=0.78\textwidth]{fig_universal_level_cycle_counterexample.png} -\caption{The $8$-vertex counterexample to the universal-source form. -With source $S=\{7\}$, the level cycle $(3,4,5,8)$ lies in $L_2$ and -forces all four colours in every proper $4$-vertex-colouring.} -\label{fig:universal-level-cycle-counterexample} -\end{figure} - -\begin{example}[Counterexample to Conjecture~\ref{conj:false-universal-level-cycle-three-colour}] -\label{ex:universal-level-cycle-counterexample} -Let $G$ be the maximal planar graph on vertex set -$\{1,2,3,4,5,6,7,8\}$ with edge set -\[ -\begin{aligned} -E(G)=\{& -12,13,14,15,16,17,23,26,27,34,35,36,38,\\ -&45,56,58,67,68\}. -\end{aligned} -\] -Here $ij$ denotes the edge $\{i,j\}$. -Take the vertex source $S=\{7\}$. The corresponding levels are -\[ -L_0=\{7\},\qquad L_1=\{1,2,6\},\qquad -L_2=\{3,4,5,8\}. -\] -Inside $G[L_2]$ the vertices $(3,4,5,8)$ form a simple cycle. In -every proper $4$-vertex-colouring of $G$, these four vertices receive -four distinct colours. The edges $34$, $45$, $58$, $38$, and $35$ -force all pairs among $\{3,4,5,8\}$ except possibly $\{4,8\}$ to have -distinct colours. If $4$ and $8$ had the same colour, then vertex $6$, -which is adjacent to $3$, $5$, and $8$, would have to use the fourth -colour; but vertex $1$ is adjacent to $3$, $4$, $5$, and $6$, and -would then be adjacent to all four colours, impossible in a proper -$4$-colouring. Hence $4$ and $8$ also have distinct colours, so the -level cycle $(3,4,5,8)$ uses all four colours in every proper -$4$-colouring of $G$. Therefore no proper $4$-colouring has the -level-cycle three-colour restriction with respect to $S=\{7\}$. -\end{example} - -\subsection*{An inner-boundary refinement} - -The level-cycle restriction constrains \emph{every} simple cycle in -every level. For the tire-tree program, the cycles that actually carry -boundary state are fewer: each tire transfers colour information across -its tread between its two boundaries -(Theorem~\ref{thm:tire-chromatic-polynomial-transfer}), so it is the tire -\emph{inner boundaries} $B_{\mathrm{in}}^{(T)}$ --- not all level cycles ---- that one wishes to compress. This motivates a restriction stated -directly in the objects of the decomposition. - -\begin{definition}[Tire inner-boundary three-colour restriction] -\label{def:tire-inner-boundary-three-colour} -Let $G$ be a maximal planar graph, let $v_0 \in V(G)$ be a vertex source -on the outer face of $\Pi_G$, and let $c \colon V(G) \to \{1,2,3,4\}$ be -a proper $4$-vertex-colouring of $G$. We say $c$ has the \emph{tire -inner-boundary three-colour restriction} with respect to -$\mathcal{T}(G, \{v_0\})$ if every tire tread $T \in -\mathcal{T}(G, \{v_0\})$ satisfies -\[ - |c(V(B_{\mathrm{in}}^{(T)}))| \leq 3, -\] -i.e.\ the inner boundary of every tire omits at least one of the four -colours. (A degenerate inner boundary is a single vertex and the -condition is then vacuous.) -\end{definition} - -\begin{conjecture}[Tire inner-boundary three-colour conjecture] -\label{conj:tire-inner-boundary-three-colour} -Every maximal planar graph $G$ admits a vertex source $v_0 \in V(G)$ and -a proper $4$-vertex-colouring $c$ of $G$ such that $c$ has the tire -inner-boundary three-colour restriction with respect to -$\mathcal{T}(G, \{v_0\})$. -\end{conjecture} - -\subsection*{A counterexample at $n=14$} - -Conjecture~\ref{conj:tire-inner-boundary-three-colour} is in fact -false. An exhaustive search over the triangulations enumerated by -\texttt{plantri} at $n=14$ encounters a graph $G^\star$ on $14$ vertices -and $36$ edges --- specifically, the graph at index $263993$ in the -\texttt{plantri} enumeration --- for which no vertex source admits any -witness. - -\begin{example}[Counterexample to Conjecture~\ref{conj:tire-inner-boundary-three-colour}] -\label{ex:inner-boundary-counterexample} -Let $G^\star$ be the maximal planar graph with vertex set -$\{1,2,\dots,14\}$ and edge set -\begin{align*} -E(G^\star) = \{ - & 12, 13, 14, 15, 16, 17, 18, \\ - & 23, 24, 26, 28, 29, 2\,10, \\ - & 34, 45, 46, 4\,10, 56, 67, 69, 6\,10, \\ - & 78, 79, 7\,11, 7\,12, 7\,13, \\ - & 89, 8\,12, 8\,13, 8\,14, \\ - & 9\,11, 9\,12, 9\,14, \\ - & 11\,12, 12\,13, 12\,14 -\}. -\end{align*} -The graph $G^\star$ is a $3$-connected (but not $5$-connected) planar -triangulation with degree sequence -$(7,7,7,7,7,7,6,6,3,3,3,3,3,3)$ and exactly $96$ proper $4$-vertex -colourings. For \emph{every} choice of vertex source -$v_0 \in V(G^\star)$, each of the $96$ proper $4$-colourings of -$G^\star$ has some tire whose inner boundary uses all four colours. -A planar embedding is shown in -Figure~\ref{fig:inner-boundary-counterexample}. -\end{example} - -\begin{figure}[ht] -\centering -\includegraphics[width=0.78\textwidth]{fig_inner_boundary_counterexample} -\caption{The $14$-vertex counterexample $G^\star$ to -Conjecture~\ref{conj:tire-inner-boundary-three-colour} in a planar -embedding. The six degree-$3$ vertices split into two triples, -$\{3,5,10\}$ each adjacent to a triangle in the -core $\{1,2,4,6\}$, and $\{11,13,14\}$ each adjacent to a triangle in -the core $\{7,8,9,12\}$; the two cores are joined by the edges -$17,28,69$ together with $12$.} -\label{fig:inner-boundary-counterexample} -\end{figure} - -The failure was verified by enumerating, for each of the $14$ vertex -sources, all $96$ proper $4$-colourings of $G^\star$ and computing the -inner boundary $V(B_{\mathrm{in}}^{(T)})$ of every tire $T$ as the -level-$(d+1)$ vertices of the corresponding depth-$d$ dual component. -Each source has exactly two non-degenerate inner boundaries -(size $\geq 4$), and every proper $4$-colouring assigns all four -colours to at least one of them. - -The graph $G^\star$ does not refute -Conjecture~\ref{conj:level-cycle-three-colour}: the vertex source -$v_0 = 10$ admits a proper $4$-colouring under which every simple level -cycle uses at most three colours. - -\subsection*{The surviving level-cycle conjecture} - -\begin{conjecture}[Level-cycle three-colour conjecture] -\label{conj:level-cycle-three-colour} -Let $G$ be a maximal planar graph. Then there exists a level source -$S \subseteq V(G)$ such that $G$ admits a proper $4$-vertex-colouring -with the level-cycle three-colour restriction with respect to $S$. -\end{conjecture} - -\subsection*{Enumeration for small $n$} - -We exhaustively enumerated all plane triangulation isomorphism classes with -$4 \leq n \leq 13$ vertices and searched the vertex sources for each graph. -No counterexample to Conjecture~\ref{conj:level-cycle-three-colour} appeared -in this range. Table~\ref{tab:level-cycle-three-colour-counts} records the -size of the search space and the number of triangulations that admit a -witness. - -\begin{table}[ht] -\centering -\small -\setlength{\tabcolsep}{4pt} -\begin{tabular}{ccc} -$n$ & triangulations & with witness \\\hline -$4$ & $1$ & $1$ \\ -$5$ & $1$ & $1$ \\ -$6$ & $2$ & $2$ \\ -$7$ & $5$ & $5$ \\ -$8$ & $14$ & $14$ \\ -$9$ & $50$ & $50$ \\ -$10$ & $233$ & $233$ \\ -$11$ & $1249$ & $1249$ \\ -$12$ & $7595$ & $7595$ \\ -$13$ & $49566$ & $49566$ \\ -\end{tabular} -\caption{Exhaustive vertex-source search for the level-cycle three-colour conjecture on all triangulation isomorphism classes with $4 \leq n \leq 13$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.} -\label{tab:level-cycle-three-colour-counts} -\end{table} - -We also tested the six dual triangulations of the Holton--McKay graphs, -which lie just beyond this census, and found witnesses in each case. - -\subsection*{The $5$-connected slice at $n \leq 24$} - -As a compact test above the full small-$n$ census, we also enumerated the -$5$-connected triangulations at $14 \leq n \leq 24$ with \texttt{plantri --c5 -a}. These are especially rigid triangulations, and the slice remains -small enough to check exhaustively. Every graph in this slice admits a -vertex source witnessing Conjecture~\ref{conj:level-cycle-three-colour}. - -\begin{table}[ht] -\centering -\small -\setlength{\tabcolsep}{4pt} -\begin{tabular}{ccc} -$n$ & $5$-connected triangulations & with witness \\\hline -$14$ & $1$ & $1$ \\ -$15$ & $1$ & $1$ \\ -$16$ & $3$ & $3$ \\ -$17$ & $4$ & $4$ \\ -$18$ & $12$ & $12$ \\ -$19$ & $23$ & $23$ \\ -$20$ & $71$ & $71$ \\ -$21$ & $187$ & $187$ \\ -$22$ & $627$ & $627$ \\ -$23$ & $1970$ & $1970$ \\ -$24$ & $6833$ & $6833$ \\ -\end{tabular} -\caption{The $5$-connected triangulations at $14 \leq n \leq 24$ generated by -\texttt{plantri -c5 -a}. All $9732$ graphs in this slice admit a vertex -source witnessing the level-cycle three-colour conjecture.} -\label{tab:level-cycle-three-colour-c5-14-16} -\end{table} - -\begin{definition}[Seam] -\label{def:seam} -A \emph{seam} of a maximal planar graph $G$ is a simple cycle -$C \subset G$ such that, for some vertex $v_0 \in V(G)$, $C = -B_{\mathrm{out}}^{(T)}$ for some non-root tread $T$ in -$\mathcal{T}(G, \{v_0\})$. - -By Theorem~\ref{thm:tire-tree-decomposition}, every seam $C$ separates -$G$ into: -\begin{itemize} -\item the \emph{seam interior} $G_T$, the triangulated disk on the - $T$-descendant side of $C$; -\item the \emph{seam exterior} $G_C^{\mathrm{ext}} := G \setminus - \mathrm{int}(G_T)$, the triangulated polygon with outer face - bounded by $C$ on the side containing $v_0$; -\end{itemize} -both sharing $C$. A seam is \emph{non-trivial} if both -$V(G_T) \setminus V(C)$ and $V(G_C^{\mathrm{ext}}) \setminus V(C)$ are -non-empty. - -For any seam $C$ and either side $X \in \{G_T, G_C^{\mathrm{ext}}\}$, -write -\[ - \mathrm{Col}(X \mid C) \;:=\; \bigl\{\, c|_{V(C)} \;:\; c \text{ a - proper $4$-colouring of } X \,\bigr\} \;\subseteq\; \{1,2,3,4\}^{V(C)} -\] -for the set of $C$-restricted $4$-colourings induced by $4$-colourings -of $X$ (each element is a proper $4$-colouring of the cycle $C$). -\end{definition} - -\begin{definition}[Partial tire tree] -\label{def:partial-tire-tree} -Let $T_r$ be a tire tread in $\mathcal{T}(G, S)$ with outer boundary -cycle $C_{T_r} = B_{\mathrm{out}}^{(T_r)}$, and let $G_{T_r}$ be the -triangulated disk inside $C_{T_r}$ given by -Theorem~\ref{thm:tire-tree-decomposition}. The \emph{partial tire -tree} with root $T_r$, written $G_{T_r}^{\circ}$, is the induced -subgraph of $G$ on the vertex set -$V(G_{T_r}) \setminus V(C_{T_r})$ --- -i.e.\ $G_{T_r}$ with the seam-cycle vertices removed. - -Equivalently, $V(G_{T_r}^{\circ})$ is the set of vertices of $G$ -strictly inside $C_{T_r}$ on the side away from the level source, -and $E(G_{T_r}^{\circ})$ consists of the edges of $G$ both of whose -endpoints lie in this strict interior. The tree-of-tire-treads -structure of $G_{T_r}^{\circ}$ is the sub-tree of $\mathcal{T}(G, S)$ -rooted at $T_r$, with $T_r$'s outer boundary peeled away. -\end{definition} - -\begin{lemma}[Seam edges are shared by at most one other depth-$d$ seam] -\label{lem:seam-edge-shared} -Let $G$ be a maximal planar graph with single-vertex level source -$S = \{v_0\}$, fix $d \ge 1$, and let $e \in E(G)$ be an edge lying on -the seam $C_T = B_{\mathrm{out}}^{(T)}$ of some tire tread -$T \in \mathcal{T}(G, S)$ at depth $d$. Then there is at most one -other tire tread $T' \in \mathcal{T}(G, S)$ at the same depth $d$ with -$e \in C_{T'}$. -\end{lemma} - -\begin{proof} -By Theorem~\ref{thm:tread-tree}, $C_T$ is the boundary cycle of a -bounded face of the parent's inner outerplanar graph $O^{(T_p)}$, -where $T_p \in \mathcal{T}(G, S)$ is the parent of $T$ at depth -$d - 1$. The inner dual of a connected outerplanar graph is a tree, -so each edge of -$O^{(T_p)}$ lies on at most two of its bounded face cycles. Hence -$e$ lies on at most one other bounded face cycle of $O^{(T_p)}$, -corresponding (Theorem~\ref{thm:tread-tree}, child--face bijection) -to at most one sibling of $T$ at depth $d$ whose seam contains $e$. -\end{proof} - -\begin{conjecture}[Seam structure of minimum $4$CT counterexamples, sketch] -\label{conj:seam-counterexample} -Suppose the Four Colour Theorem fails: there exists a maximal planar -graph that is not $4$-colourable. Let $G$ be a \emph{minimum} such -counterexample (with $|V(G)|$ minimal among non-$4$-colourable maximal -planar graphs). Then: - -\medskip - -\noindent\emph{Restatement-of-classical content.} -\begin{itemize} -\item[(C1)] \emph{Bilateral colourability.} For every non-trivial seam - $C$ of $G$, both $\mathrm{Col}(G_T \mid C)$ and - $\mathrm{Col}(G_C^{\mathrm{ext}} \mid C)$ are non-empty. -\item[(C2)] \emph{Bilateral incompatibility.} For every non-trivial - seam $C$, - \[ - \mathrm{Col}(G_T \mid C) \;\cap\; - \mathrm{Col}(G_C^{\mathrm{ext}} \mid C) \;=\; \emptyset. - \] -\item[(C3)] \emph{Length lower bound (Birkhoff).} Every non-trivial - seam $C$ of $G$ has $|V(C)| \ge 6$. -\end{itemize} - -(C1) and (C2) together restate ``$G$ is a counterexample whose every -internal cut by a seam splits into two colourable pieces with -incompatible boundary palettes''; (C1) follows from minimality applied -to each side after closing the polygonal outer face by a single apex, -(C2) from $G$ itself being non-$4$-colourable. (C3) is Birkhoff's -internally-$6$-connected condition restated in the seam language. - -\medskip - -\noindent\emph{Substantive (speculative) content.} -\begin{itemize} -\item[(C4)] \emph{Innermost obstruction.} There exists a vertex source - $v_0 \in V(G)$ and a \emph{leaf} tread $T^* \in - \mathcal{T}(G, \{v_0\})$ (a tread with no children in the - tree-of-treads) such that: - \begin{enumerate} - \item[(i)] the seam interior $G_{T^*}$ is, up to plane - iso, one of a finite list of \emph{minimal seam - configurations}, characterized by their boundary - palette $\mathrm{Col}(G_{T^*} \mid C_{T^*})$ being a - specific proper subset of the proper $4$-colourings - of the cycle $C_{T^*}$; - \item[(ii)] the path in $\mathcal{T}(G, \{v_0\})$ from the - root $T_0$ to $T^*$ is an \emph{obstruction chain}: - $\mathrm{Col}(G_T \mid C_T)$ is monotonically - restricted (under the natural pull-back along - parent--child seams of - Remark~\ref{rem:tree-coloring-factorisation}) as $T$ - descends from the root to $T^*$, with the final - restriction at $T^*$ being incompatible with the - $v_0$-side palette. - \end{enumerate} -\end{itemize} -\end{conjecture} - \begin{thebibliography}{9} \bibitem{tait-original} diff --git a/papers/three_color_restrictions_nested_tire_graphs/fig_inner_boundary_counterexample.png b/papers/three_color_restrictions_nested_tire_graphs/fig_inner_boundary_counterexample.png new file mode 100644 index 0000000..11da609 Binary files /dev/null and b/papers/three_color_restrictions_nested_tire_graphs/fig_inner_boundary_counterexample.png differ diff --git a/papers/three_color_restrictions_nested_tire_graphs/fig_universal_level_cycle_counterexample.png b/papers/three_color_restrictions_nested_tire_graphs/fig_universal_level_cycle_counterexample.png new file mode 100644 index 0000000..89ebd2d Binary files /dev/null and b/papers/three_color_restrictions_nested_tire_graphs/fig_universal_level_cycle_counterexample.png differ diff --git a/papers/three_color_restrictions_nested_tire_graphs/paper.aux b/papers/three_color_restrictions_nested_tire_graphs/paper.aux new file mode 100644 index 0000000..4a149ec --- /dev/null +++ b/papers/three_color_restrictions_nested_tire_graphs/paper.aux @@ -0,0 +1,64 @@ +\relax +\citation{bauerfeld-nested-tire-decompositions} +\citation{bauerfeld-nested-tire-duals} +\citation{birkhoff-reducibility} +\citation{birkhoff-lewis-chromatic} +\citation{tutte-chromatic-sums-1973} +\citation{tutte-algebraic-colorings} +\citation{tutte-four-colour-conjecture} +\citation{dvorak-lidicky-cones} +\citation{heesch-untersuchungen} +\citation{robertson-sanders-seymour-thomas} +\citation{robertson-sanders-seymour-thomas} +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Related work.}}{1}{}\protected@file@percent } +\citation{bauerfeld-nested-tire-decompositions} +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Background from nested tire decompositions}}{2}{}\protected@file@percent } +\newlabel{rem:level-cycle-motivation}{{2.1}{2}} +\newlabel{def:level-cycle-three-colour-restriction}{{2.2}{2}} +\newlabel{conj:false-universal-level-cycle-three-colour}{{2.3}{2}} +\newlabel{ex:universal-level-cycle-counterexample}{{2.4}{2}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The $8$-vertex counterexample to the universal-source form. With source $S=\{7\}$, the level cycle $(3,4,5,8)$ lies in $L_2$ and forces all four colours in every proper $4$-vertex-colouring.}}{3}{}\protected@file@percent } +\newlabel{fig:universal-level-cycle-counterexample}{{1}{3}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{An inner-boundary refinement}}{3}{}\protected@file@percent } +\newlabel{def:tire-inner-boundary-three-colour}{{2.5}{3}} +\newlabel{conj:tire-inner-boundary-three-colour}{{2.6}{4}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A counterexample at $n=14$}}{4}{}\protected@file@percent } +\newlabel{ex:inner-boundary-counterexample}{{2.7}{4}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The $14$-vertex counterexample $G^\star $ to Conjecture\nonbreakingspace 2.6\hbox {} in a planar embedding. The six degree-$3$ vertices split into two triples, $\{3,5,10\}$ each adjacent to a triangle in the core $\{1,2,4,6\}$, and $\{11,13,14\}$ each adjacent to a triangle in the core $\{7,8,9,12\}$; the two cores are joined by the edges $17,28,69$ together with $12$.}}{4}{}\protected@file@percent } +\newlabel{fig:inner-boundary-counterexample}{{2}{4}} +\citation{bauerfeld-nested-tire-decompositions} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The surviving level-cycle conjecture}}{5}{}\protected@file@percent } +\newlabel{conj:level-cycle-three-colour}{{2.8}{5}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{5}{}\protected@file@percent } +\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Exhaustive vertex-source search for the level-cycle three-colour conjecture on all triangulation isomorphism classes with $4 \leq n \leq 13$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.}}{5}{}\protected@file@percent } +\newlabel{tab:level-cycle-three-colour-counts}{{1}{5}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{5}{}\protected@file@percent } +\newlabel{def:seam}{{2.9}{5}} +\citation{bauerfeld-nested-tire-decompositions} +\citation{bauerfeld-nested-tire-decompositions} +\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The $5$-connected triangulations at $14 \leq n \leq 24$ generated by \texttt {plantri -c5 -a}. All $9732$ graphs in this slice admit a vertex source witnessing the level-cycle three-colour conjecture.}}{6}{}\protected@file@percent } +\newlabel{tab:level-cycle-three-colour-c5-14-16}{{2}{6}} +\newlabel{def:partial-tire-tree}{{2.10}{6}} +\newlabel{lem:seam-edge-shared}{{2.11}{6}} +\citation{bauerfeld-nested-tire-decompositions} +\bibcite{tait-original}{1} +\bibcite{bauerfeld-depth}{2} +\bibcite{bauerfeld-nested-tire-decompositions}{3} +\bibcite{bauerfeld-nested-tire-duals}{4} +\bibcite{birkhoff-reducibility}{5} +\bibcite{birkhoff-lewis-chromatic}{6} +\bibcite{tutte-four-colour-conjecture}{7} +\bibcite{tutte-algebraic-colorings}{8} +\bibcite{tutte-chromatic-sums-1973}{9} +\bibcite{heesch-untersuchungen}{10} +\bibcite{robertson-sanders-seymour-thomas}{11} +\bibcite{dvorak-lidicky-cones}{12} +\newlabel{tocindent-1}{0pt} +\newlabel{tocindent0}{14.69437pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{0pt} +\newlabel{tocindent3}{0pt} +\newlabel{conj:seam-counterexample}{{2.12}{7}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent } +\gdef \@abspage@last{8} diff --git a/papers/three_color_restrictions_nested_tire_graphs/paper.fdb_latexmk b/papers/three_color_restrictions_nested_tire_graphs/paper.fdb_latexmk new file mode 100644 index 0000000..8e57c2d --- /dev/null +++ b/papers/three_color_restrictions_nested_tire_graphs/paper.fdb_latexmk @@ -0,0 +1,145 @@ +# Fdb version 3 +["pdflatex"] 1780944697 "paper.tex" "paper.pdf" "paper" 1780944698 + "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex9.tfm" 1246382020 996 a18840b13b499c08ac2de96a99eda4bc "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1246382020 924 9904cf1d39e9767e7a3622f2a125a565 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1246382020 908 2921f8a10601f252058503cc6570e581 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1246382020 940 75ac932a52f80982a9f8ea75d03a34cf "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm" 1136768653 1328 c834bbb027764024c09d3d2bf908b5f0 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm" 1136768653 1332 1fde11373e221473104d6cc5993f046e "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm" 1136768653 1300 63a6111ee6274895728663cf4b4e7e81 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm" 1136768653 1512 f21f83efb36853c0b70002322c1ab3ad "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm" 1136768653 1520 eccf95517727cb11801f4f1aee3a21b4 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi9.tfm" 1136768653 1524 d89e2d087a9828407a196f428428ef4a "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm" 1136768653 1300 b62933e007d01cfd073f79b963c01526 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr9.tfm" 1136768653 1292 6b21b9c2c7bebb38aa2273f7ca0fb3af "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy9.tfm" 1136768653 1116 25a7bf822c58caf309a702ef79f4afbb "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm" 1136768653 768 1321e9409b4137d6fb428ac9dc956269 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb" 1248133631 32166 b0c356b15f19587482a9217ce1d8fa67 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb" 1248133631 37912 77d683123f92148345f3fc36a38d9ab1 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1248133631 35469 70d41d2b9ea31d5d813066df7c99281c "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi9.pfb" 1248133631 36094 798f80770b3b148ceedd006d487db67c "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr9.pfb" 1248133631 33993 9b89b85fd2d9df0482bd47194d1d3bf3 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb" 1248133631 32915 7bf7720c61a5b3a7ff25b0964421c9b6 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb" 1248133631 35660 fb24af7afbadb71801619f1415838111 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb" 1248133631 31099 c85edf1dd5b9e826d67c9c7293b6786c "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 "" + "/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex" 1601326656 992 855ff26741653ab54814101ca36e153c "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex" 1601326656 43820 1fef971b75380574ab35a0d37fd92608 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex" 1601326656 19324 f4e4c6403dd0f1605fd20ed22fa79dea "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex" 1601326656 6038 ccb406740cc3f03bbfb58ad504fe8c27 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex" 1601326656 6944 e12f8f7a7364ddf66f93ba30fb3a3742 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex" 1601326656 4883 42daaf41e27c3735286e23e48d2d7af9 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex" 1601326656 2544 8c06d2a7f0f469616ac9e13db6d2f842 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex" 1601326656 44195 5e390c414de027626ca5e2df888fa68d "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex" 1601326656 17311 2ef6b2e29e2fc6a2fc8d6d652176e257 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex" 1601326656 21302 788a79944eb22192a4929e46963a3067 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex" 1601326656 9690 01feb7cde25d4293ef36eef45123eb80 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex" 1601326656 33335 dd1fa4814d4e51f18be97d88bf0da60c "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex" 1601326656 2965 4c2b1f4e0826925746439038172e5d6f "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex" 1601326656 5196 2cc249e0ee7e03da5f5f6589257b1e5b "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex" 1601326656 20726 d4c8db1e2e53b72721d29916314a22ea "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex" 1601326656 35249 abd4adf948f960299a4b3d27c5dddf46 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex" 1601326656 21989 fdc867d05d228316de137a9fc5ec3bbe "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex" 1601326656 8893 e851de2175338fdf7c17f3e091d94618 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex" 1601326656 4572 4a19637ef65ce88ad2f2d5064b69541d "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex" 1608933718 11518 738408f795261b70ce8dd47459171309 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex" 1621110968 186007 6e7dfe0bd57520fd5f91641aa72dcac8 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex" 1601326656 32995 ac577023e12c0e4bd8aa420b2e852d1a "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex" 1557692582 3063 8c415c68a0f3394e45cfeca0b65f6ee6 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex" 1601326656 521 8e224a7af69b7fee4451d1bf76b46654 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex" 1601326656 13391 84d29568c13bdce4133ab4a214711112 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex" 1601326656 104935 184ed87524e76d4957860df4ce0cd1c3 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex" 1601326656 10165 cec5fa73d49da442e56efc2d605ef154 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex" 1601326656 28178 41c17713108e0795aac6fef3d275fbca "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex" 1601326656 9989 c55967bf45126ff9b061fa2ca0c4694f "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex" 1601326656 3865 ac538ab80c5cf82b345016e474786549 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex" 1557692582 3177 27d85c44fbfe09ff3b2cf2879e3ea434 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex" 1621110968 11024 0179538121bc2dba172013a3ef89519f "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex" 1608933718 7854 4176998eeefd8745ac6d2d4bd9c98451 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex" 1601326656 3379 781797a101f647bab82741a99944a229 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex" 1601326656 92405 f515f31275db273f97b9d8f52e1b0736 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex" 1601326656 37376 11cd75aac3da1c1b152b2848f30adc14 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex" 1601326656 8471 c2883569d03f69e8e1cabfef4999cfd7 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex" 1601326656 21201 08d231a2386e2b61d64641c50dc15abd "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex" 1601326656 16121 346f9013d34804439f7436ff6786cef7 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex" 1621110968 44784 cedaa399d15f95e68e22906e2cc09ef8 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex" 1621110968 465 d68603f8b820ea4a08cce534944db581 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg" 1601326656 926 2963ea0dcf6cc6c0a770b69ec46a477b "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def" 1601326656 5546 f3f24d7898386cb7daac70bdd2c4d6dc "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def" 1601326656 12601 4786e597516eddd82097506db7cfa098 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex" 1621110968 61163 9b2eefc24e021323e0fc140e9826d016 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex" 1601326656 1896 b8e0ca0ac371d74c0ca05583f6313c91 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex" 1601326656 7778 53c8b5623d80238f6a20aa1df1868e63 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex" 1606168878 23997 a4bed72405fa644418bea7eac2887006 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex" 1621110968 37060 797782f0eb50075c9bc952374d9a659a "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex" 1601326656 37431 9abe862035de1b29c7a677f3205e3d9f "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex" 1601326656 4494 af17fb7efeafe423710479858e42fa7e "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex" 1601326656 7251 fb18c67117e09c64de82267e12cd8aa4 "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex" 1621110968 29274 e15c5b7157d21523bd9c9f1dfa146b8e "" + "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def" 1621110968 6825 a2b0ea5b539dda0625e99dd15785ab59 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls" 1591045760 61881 a7369c346c2922a758ae6283cc1ed014 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty" 1359763108 13829 94730e64147574077f8ecfea9bb69af4 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd" 1359763108 961 6518c6525a34feb5e8250ffa91731cff "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd" 1359763108 961 d02606146ba5601b5645f987c92e6193 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1622667781 2222 da905dc1db75412efd2d8f67739f0596 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty" 1622667781 4173 bc0410bcccdff806d6132d3c1ef35481 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty" 1636758526 87648 07fbb6e9169e00cb2a2f40b31b2dbf3c "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty" 1636758526 4128 8eea906621b6639f7ba476a472036bbe "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty" 1636758526 2444 926f379cc60fcf0c6e3fee2223b4370d "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1579991033 13886 d1306dcf79a944f6988e688c1785f9ce "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg" 1459978653 1213 620bba36b25224fa9b7e1ccb4ecb76fd "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg" 1465944070 1224 978390e9c2234eab29404bc21b268d1e "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def" 1601931164 19103 48d29b6e2a64cb717117ef65f107b404 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty" 1622581934 18399 7e40f80366dffb22c0e7b70517db5cb4 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty" 1636758526 7996 a8fb260d598dcaf305a7ae7b9c3e3229 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty" 1622581934 2671 4de6781a30211fe0ea4c672e4a2a8166 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty" 1636758526 4009 187ea2dc3194cd5a76cd99a8d7a6c4d0 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def" 1644269979 29921 d0acc05a38bd4aa3af2017f0b7c137ce "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty" 1601326656 1090 bae35ef70b3168089ef166db3e66f5b2 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty" 1601326656 410 615550c46f918fcbee37641b02a862d9 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty" 1601326656 21013 f4ff83d25bb56552493b030f27c075ae "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty" 1601326656 989 c49c8ae06d96f8b15869da7428047b1e "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty" 1601326656 339 c2e180022e3afdb99c7d0ea5ce469b7d "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty" 1601326656 306 c56a323ca5bf9242f54474ced10fca71 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty" 1601326656 443 8c872229db56122037e86bcda49e14f3 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty" 1601326656 348 ee405e64380c11319f0e249fed57e6c5 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty" 1601326656 274 5ae372b7df79135d240456a1c6f2cf9a "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty" 1601326656 325 f9f16d12354225b7dd52a3321f085955 "" + "/usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty" 1635798903 56029 3f7889dab51d620aa43177c391b7b190 "" + "/usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf" 1646502317 40171 cdab547de63d26590bebb3baff566530 "" + "/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 "" + "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" + "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" + "fig_inner_boundary_counterexample.png" 1780944385 86866 e15e4311e42fec20179ac6bb90683dea "" + "fig_universal_level_cycle_counterexample.png" 1780944385 75145 08f600be4e05c11d702bee45996ca222 "" + "paper.aux" 1780944698 4645 3a869b72856c307c3826ab1685ff7923 "pdflatex" + "paper.tex" 1780944666 24042 8f637a21193743241488dc904c5838c7 "" + (generated) + "paper.aux" + "paper.log" + "paper.pdf" diff --git a/papers/three_color_restrictions_nested_tire_graphs/paper.fls b/papers/three_color_restrictions_nested_tire_graphs/paper.fls new file mode 100644 index 0000000..e751d01 --- /dev/null +++ b/papers/three_color_restrictions_nested_tire_graphs/paper.fls @@ -0,0 +1,489 @@ +PWD /Users/didericis/Code/math-research/papers/three_color_restrictions_nested_tire_graphs +INPUT /usr/local/texlive/2022/texmf.cnf +INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf +INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt +INPUT paper.tex +OUTPUT paper.log +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT ./paper.aux +INPUT paper.aux +INPUT paper.aux +OUTPUT paper.aux +INPUT /usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm +INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii +INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii +INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii +INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm +OUTPUT paper.pdf +INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map +INPUT ./fig_universal_level_cycle_counterexample.png +INPUT ./fig_universal_level_cycle_counterexample.png +INPUT fig_universal_level_cycle_counterexample.png +INPUT ./fig_universal_level_cycle_counterexample.png +INPUT ./fig_universal_level_cycle_counterexample.png +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm +INPUT ./fig_inner_boundary_counterexample.png +INPUT fig_inner_boundary_counterexample.png +INPUT ./fig_inner_boundary_counterexample.png +INPUT ./fig_inner_boundary_counterexample.png +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr9.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi9.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy9.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex9.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm +INPUT paper.aux +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx8.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi9.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr9.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb diff --git a/papers/three_color_restrictions_nested_tire_graphs/paper.log b/papers/three_color_restrictions_nested_tire_graphs/paper.log new file mode 100644 index 0000000..61f8866 --- /dev/null +++ b/papers/three_color_restrictions_nested_tire_graphs/paper.log @@ -0,0 +1,555 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 8 JUN 2026 14:51 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**paper.tex +(./paper.tex +LaTeX2e <2021-11-15> patch level 1 +L3 programming layer <2022-02-24> +(/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +Document Class: amsart 2020/05/29 v2.20.6 +\linespacing=\dimen138 +\normalparindent=\dimen139 +\normaltopskip=\skip47 +(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2021/10/15 v2.17l AMS math features +\@mathmargin=\skip48 + +For additional information on amsmath, use the `?' option. +(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2021/08/26 v2.01 AMS text + +(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 generic functions +\@emptytoks=\toks16 +\ex@=\dimen140 +)) +(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d Bold Symbols +\pmbraise@=\dimen141 +) +(/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 2021/08/26 v2.02 operator names +) +\inf@bad=\count185 +LaTeX Info: Redefining \frac on input line 234. +\uproot@=\count186 +\leftroot@=\count187 +LaTeX Info: Redefining \overline on input line 399. +\classnum@=\count188 +\DOTSCASE@=\count189 +LaTeX Info: Redefining \ldots on input line 496. +LaTeX Info: Redefining \dots on input line 499. +LaTeX Info: Redefining \cdots on input line 620. +\Mathstrutbox@=\box50 +\strutbox@=\box51 +\big@size=\dimen142 +LaTeX Font Info: Redeclaring font encoding OML on input line 743. +LaTeX Font Info: Redeclaring font encoding OMS on input line 744. +\macc@depth=\count190 +\c@MaxMatrixCols=\count191 +\dotsspace@=\muskip16 +\c@parentequation=\count192 +\dspbrk@lvl=\count193 +\tag@help=\toks17 +\row@=\count194 +\column@=\count195 +\maxfields@=\count196 +\andhelp@=\toks18 +\eqnshift@=\dimen143 +\alignsep@=\dimen144 +\tagshift@=\dimen145 +\tagwidth@=\dimen146 +\totwidth@=\dimen147 +\lineht@=\dimen148 +\@envbody=\toks19 +\multlinegap=\skip49 +\multlinetaggap=\skip50 +\mathdisplay@stack=\toks20 +LaTeX Info: Redefining \[ on input line 2938. +LaTeX Info: Redefining \] on input line 2939. +) +LaTeX Font Info: Trying to load font information for U+msa on input line 397 +. + +(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2013/01/14 v3.01 AMS symbols A +) +(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Redeclaring math symbol \hbar on input line 98. +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 106. +) +\copyins=\insert199 +\abstractbox=\box52 +\listisep=\skip51 +\c@part=\count197 +\c@section=\count198 +\c@subsection=\count266 +\c@subsubsection=\count267 +\c@paragraph=\count268 +\c@subparagraph=\count269 +\c@figure=\count270 +\c@table=\count271 +\abovecaptionskip=\skip52 +\belowcaptionskip=\skip53 +\captionindent=\dimen149 +\thm@style=\toks21 +\thm@bodyfont=\toks22 +\thm@headfont=\toks23 +\thm@notefont=\toks24 +\thm@headpunct=\toks25 +\thm@preskip=\skip54 +\thm@postskip=\skip55 +\thm@headsep=\skip56 +\dth@everypar=\toks26 +) +(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty +Package: amssymb 2013/01/14 v3.01 AMS font symbols +) +(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR) + +(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty +Package: keyval 2014/10/28 v1.15 key=value parser (DPC) +\KV@toks@=\toks27 +) +(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty +Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR) + +(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty +Package: trig 2021/08/11 v1.11 sin cos tan (DPC) +) +(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration +) +Package graphics Info: Driver file: pdftex.def on input line 107. + +(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def +File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex +)) +\Gin@req@height=\dimen150 +\Gin@req@width=\dimen151 +) +(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.te +x +\pgfutil@everybye=\toks28 +\pgfutil@tempdima=\dimen152 +\pgfutil@tempdimb=\dimen153 + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-li +sts.tex)) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def +\pgfutil@abb=\box53 +) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/pgf.revision.tex) +Package: pgfrcs 2021/05/15 v3.1.9a (3.1.9a) +)) +Package: pgf 2021/05/15 v3.1.9a (3.1.9a) + +(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +Package: pgfsys 2021/05/15 v3.1.9a (3.1.9a) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +\pgfkeys@pathtoks=\toks29 +\pgfkeys@temptoks=\toks30 + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.c +ode.tex +\pgfkeys@tmptoks=\toks31 +)) +\pgf@x=\dimen154 +\pgf@y=\dimen155 +\pgf@xa=\dimen156 +\pgf@ya=\dimen157 +\pgf@xb=\dimen158 +\pgf@yb=\dimen159 +\pgf@xc=\dimen160 +\pgf@yc=\dimen161 +\pgf@xd=\dimen162 +\pgf@yd=\dimen163 +\w@pgf@writea=\write3 +\r@pgf@reada=\read2 +\c@pgf@counta=\count272 +\c@pgf@countb=\count273 +\c@pgf@countc=\count274 +\c@pgf@countd=\count275 +\t@pgf@toka=\toks32 +\t@pgf@tokb=\toks33 +\t@pgf@tokc=\toks34 +\pgf@sys@id@count=\count276 + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg +File: pgf.cfg 2021/05/15 v3.1.9a (3.1.9a) +) +Driver file for pgf: pgfsys-pdftex.def + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.d +ef +File: pgfsys-pdftex.def 2021/05/15 v3.1.9a (3.1.9a) + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-p +df.def +File: pgfsys-common-pdf.def 2021/05/15 v3.1.9a (3.1.9a) +))) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath. +code.tex +File: pgfsyssoftpath.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgfsyssoftpath@smallbuffer@items=\count277 +\pgfsyssoftpath@bigbuffer@items=\count278 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol. +code.tex +File: pgfsysprotocol.code.tex 2021/05/15 v3.1.9a (3.1.9a) +)) (/usr/local/texlive/2022/texmf-dist/tex/latex/xcolor/xcolor.sty +Package: xcolor 2021/10/31 v2.13 LaTeX color extensions (UK) + +(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/color.cfg +File: color.cfg 2016/01/02 v1.6 sample color configuration +) +Package xcolor Info: Driver file: pdftex.def on input line 227. +Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1352. +Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1356. +Package xcolor Info: Model `RGB' extended on input line 1368. +Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1370. +Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1371. +Package xcolor Info: Model `tHsb' substituted by `hsb' on input line 1372. +Package xcolor Info: Model `HSB' substituted by `hsb' on input line 1373. +Package xcolor Info: Model `Gray' substituted by `gray' on input line 1374. +Package xcolor Info: Model `wave' substituted by `hsb' on input line 1375. +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +Package: pgfcore 2021/05/15 v3.1.9a (3.1.9a) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex +\pgfmath@dimen=\dimen164 +\pgfmath@count=\count279 +\pgfmath@box=\box54 +\pgfmath@toks=\toks35 +\pgfmath@stack@operand=\toks36 +\pgfmath@stack@operation=\toks37 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code. +tex +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic +.code.tex) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigo +nometric.code.tex) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.rando +m.code.tex) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.compa +rison.code.tex) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base. +code.tex) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round +.code.tex) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc. +code.tex) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integ +erarithmetics.code.tex))) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex +\c@pgfmathroundto@lastzeros=\count280 +)) (/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfint.code.tex) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.co +de.tex +File: pgfcorepoints.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgf@picminx=\dimen165 +\pgf@picmaxx=\dimen166 +\pgf@picminy=\dimen167 +\pgf@picmaxy=\dimen168 +\pgf@pathminx=\dimen169 +\pgf@pathmaxx=\dimen170 +\pgf@pathminy=\dimen171 +\pgf@pathmaxy=\dimen172 +\pgf@xx=\dimen173 +\pgf@xy=\dimen174 +\pgf@yx=\dimen175 +\pgf@yy=\dimen176 +\pgf@zx=\dimen177 +\pgf@zy=\dimen178 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconst +ruct.code.tex +File: pgfcorepathconstruct.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgf@path@lastx=\dimen179 +\pgf@path@lasty=\dimen180 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage +.code.tex +File: pgfcorepathusage.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgf@shorten@end@additional=\dimen181 +\pgf@shorten@start@additional=\dimen182 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.co +de.tex +File: pgfcorescopes.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgfpic=\box55 +\pgf@hbox=\box56 +\pgf@layerbox@main=\box57 +\pgf@picture@serial@count=\count281 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicst +ate.code.tex +File: pgfcoregraphicstate.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgflinewidth=\dimen183 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransform +ations.code.tex +File: pgfcoretransformations.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgf@pt@x=\dimen184 +\pgf@pt@y=\dimen185 +\pgf@pt@temp=\dimen186 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.cod +e.tex +File: pgfcorequick.code.tex 2021/05/15 v3.1.9a (3.1.9a) +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.c +ode.tex +File: pgfcoreobjects.code.tex 2021/05/15 v3.1.9a (3.1.9a) +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathproce +ssing.code.tex +File: pgfcorepathprocessing.code.tex 2021/05/15 v3.1.9a (3.1.9a) +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.co +de.tex +File: pgfcorearrows.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgfarrowsep=\dimen187 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.cod +e.tex +File: pgfcoreshade.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgf@max=\dimen188 +\pgf@sys@shading@range@num=\count282 +\pgf@shadingcount=\count283 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.cod +e.tex +File: pgfcoreimage.code.tex 2021/05/15 v3.1.9a (3.1.9a) + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal. +code.tex +File: pgfcoreexternal.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgfexternal@startupbox=\box58 +)) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.co +de.tex +File: pgfcorelayers.code.tex 2021/05/15 v3.1.9a (3.1.9a) +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretranspare +ncy.code.tex +File: pgfcoretransparency.code.tex 2021/05/15 v3.1.9a (3.1.9a) +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns. +code.tex +File: pgfcorepatterns.code.tex 2021/05/15 v3.1.9a (3.1.9a) +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code. +tex +File: pgfcorerdf.code.tex 2021/05/15 v3.1.9a (3.1.9a) +))) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.cod +e.tex +File: pgfmoduleshapes.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgfnodeparttextbox=\box59 +) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code. +tex +File: pgfmoduleplot.code.tex 2021/05/15 v3.1.9a (3.1.9a) +) +(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version +-0-65.sty +Package: pgfcomp-version-0-65 2021/05/15 v3.1.9a (3.1.9a) +\pgf@nodesepstart=\dimen189 +\pgf@nodesepend=\dimen190 +) +(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version +-1-18.sty +Package: pgfcomp-version-1-18 2021/05/15 v3.1.9a (3.1.9a) +)) +(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +(/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex) +) (/usr/local/texlive/2022/texmf-dist/tex/latex/pgf/math/pgfmath.sty +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex)) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +Package: pgffor 2021/05/15 v3.1.9a (3.1.9a) + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex) +\pgffor@iter=\dimen191 +\pgffor@skip=\dimen192 +\pgffor@stack=\toks38 +\pgffor@toks=\toks39 +)) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.cod +e.tex +Package: tikz 2021/05/15 v3.1.9a (3.1.9a) + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothan +dlers.code.tex +File: pgflibraryplothandlers.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgf@plot@mark@count=\count284 +\pgfplotmarksize=\dimen193 +) +\tikz@lastx=\dimen194 +\tikz@lasty=\dimen195 +\tikz@lastxsaved=\dimen196 +\tikz@lastysaved=\dimen197 +\tikz@lastmovetox=\dimen198 +\tikz@lastmovetoy=\dimen256 +\tikzleveldistance=\dimen257 +\tikzsiblingdistance=\dimen258 +\tikz@figbox=\box60 +\tikz@figbox@bg=\box61 +\tikz@tempbox=\box62 +\tikz@tempbox@bg=\box63 +\tikztreelevel=\count285 +\tikznumberofchildren=\count286 +\tikznumberofcurrentchild=\count287 +\tikz@fig@count=\count288 + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.cod +e.tex +File: pgfmodulematrix.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgfmatrixcurrentrow=\count289 +\pgfmatrixcurrentcolumn=\count290 +\pgf@matrix@numberofcolumns=\count291 +) +\tikz@expandcount=\count292 + +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/librarie +s/tikzlibrarytopaths.code.tex +File: tikzlibrarytopaths.code.tex 2021/05/15 v3.1.9a (3.1.9a) +))) +(/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/frontendlayer/tikz/librarie +s/tikzlibrarybackgrounds.code.tex +File: tikzlibrarybackgrounds.code.tex 2021/05/15 v3.1.9a (3.1.9a) +\pgf@layerbox@background=\box64 +\pgf@layerboxsaved@background=\box65 +) +\c@theorem=\count293 + +(/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX) +\l__color_backend_stack_int=\count294 +\l__pdf_internal_box=\box66 +) +(./paper.aux) +\openout1 = `paper.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 29. +LaTeX Font Info: ... okay on input line 29. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 29. +LaTeX Font Info: ... okay on input line 29. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 29. +LaTeX Font Info: ... okay on input line 29. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 29. +LaTeX Font Info: ... okay on input line 29. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 29. +LaTeX Font Info: ... okay on input line 29. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 29. +LaTeX Font Info: ... okay on input line 29. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 29. +LaTeX Font Info: ... okay on input line 29. +LaTeX Font Info: Trying to load font information for U+msa on input line 29. + + (/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2013/01/14 v3.01 AMS symbols A +) +LaTeX Font Info: Trying to load font information for U+msb on input line 29. + + +(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2013/01/14 v3.01 AMS symbols B +) +(/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii +[Loading MPS to PDF converter (version 2006.09.02).] +\scratchcounter=\count295 +\scratchdimen=\dimen259 +\scratchbox=\box67 +\nofMPsegments=\count296 +\nofMParguments=\count297 +\everyMPshowfont=\toks40 +\MPscratchCnt=\count298 +\MPscratchDim=\dimen260 +\MPnumerator=\count299 +\makeMPintoPDFobject=\count300 +\everyMPtoPDFconversion=\toks41 +) (/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf +Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 +85. + +(/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv +e +)) +[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] + +File: fig_universal_level_cycle_counterexample.png Graphic file (type png) + +Package pdftex.def Info: fig_universal_level_cycle_counterexample.png used on +input line 162. +(pdftex.def) Requested size: 280.79956pt x 156.91663pt. + +[2] [3 <./fig_universal_level_cycle_counterexample.png>] + +File: fig_inner_boundary_counterexample.png Graphic file (type png) + +Package pdftex.def Info: fig_inner_boundary_counterexample.png used on input l +ine 270. +(pdftex.def) Requested size: 280.79956pt x 156.02269pt. + [4 <./fig_inner_boundary_counterexample.png>] [5] [6] +Overfull \hbox (1.78508pt too wide) in paragraph at lines 462--464 +[]\OT1/cmr/m/n/10 Length lower bound (Birkhoff). \OT1/cmr/m/it/10 Ev-ery non-tr +ivial seam $\OML/cmm/m/it/10 C$ \OT1/cmr/m/it/10 of $\OML/cmm/m/it/10 G$ \OT1/c +mr/m/it/10 has $\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/10 (\OML/cmm/m +/it/10 C\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j ^^U + [] + +[7] [8] (./paper.aux) ) +Here is how much of TeX's memory you used: + 13582 strings out of 478268 + 271682 string characters out of 5846347 + 540742 words of memory out of 5000000 + 31407 multiletter control sequences out of 15000+600000 + 477688 words of font info for 61 fonts, out of 8000000 for 9000 + 1302 hyphenation exceptions out of 8191 + 84i,9n,89p,833b,289s stack positions out of 10000i,1000n,20000p,200000b,200000s +< +/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb> +Output written on paper.pdf (8 pages, 387544 bytes). +PDF statistics: + 139 PDF objects out of 1000 (max. 8388607) + 84 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 23 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/papers/three_color_restrictions_nested_tire_graphs/paper.pdf b/papers/three_color_restrictions_nested_tire_graphs/paper.pdf new file mode 100644 index 0000000..bf5badc Binary files /dev/null and b/papers/three_color_restrictions_nested_tire_graphs/paper.pdf differ diff --git a/papers/three_color_restrictions_nested_tire_graphs/paper.tex b/papers/three_color_restrictions_nested_tire_graphs/paper.tex new file mode 100644 index 0000000..31619ba --- /dev/null +++ b/papers/three_color_restrictions_nested_tire_graphs/paper.tex @@ -0,0 +1,566 @@ +%% filename: amsart-template.tex +%% American Mathematical Society +%% AMS-LaTeX v.2 template for use with amsart +%% ==================================================================== + +\documentclass{amsart} + +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{tikz} +\usetikzlibrary{backgrounds} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{conjecture}[theorem]{Conjecture} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\newtheorem{xca}[theorem]{Exercise} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\numberwithin{equation}{section} + +\begin{document} + +\title{Three-Colour Restrictions for Nested Tire Graphs} + +% author one information +\author{Eric Bauerfeld} +\address{} +\curraddr{} +\email{} +\thanks{} + +\subjclass[2010]{Primary } + +\keywords{plane graph, triangulation, plane depth, level edge, dual graph, tire graph} + +\date{} + +\dedicatory{} + +\begin{abstract} +We study three-colour boundary restrictions suggested by the nested +tire decomposition of a plane triangulation. A level source induces a +rooted tree of tire treads, and global colouring questions factor +through local tread colourings together with compatibility along nested +boundary cycles. We formulate a level-cycle three-colour restriction, +exhibit counterexamples to two overly strong forms, and record +exhaustive evidence for a surviving source-dependent conjecture. We +also introduce seam language for minimum Four Colour Theorem +counterexamples in the tire-tree framework. +\end{abstract} + +\maketitle + +\section{Introduction} + +A classical theorem of Tait recasts the Four Colour Theorem in dual, +edge-colouring terms: a plane triangulation $G$ is properly $4$-vertex-colourable +if and only if its dual cubic graph $G'$ is properly $3$-edge-colourable. Thus a +minimal counterexample to the Four Colour Theorem -- a smallest triangulation +admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph +admitting no proper $3$-edge-colouring. + +The structural study of such a minimal counterexample is the +overarching motivation for the present line of work. The companion +decomposition paper~\cite{bauerfeld-nested-tire-decompositions} +establishes the foundational vocabulary --- level sources, dual depth, +tire graphs, medial tire graphs, and tire-tree decompositions --- on +which this paper builds. The companion dual paper +\cite{bauerfeld-nested-tire-duals} develops nested-cycle structure +theorems and chain-pigeonhole conjectures for tire annular subgraphs +of $G'$. + +\paragraph{Related work.} +The structural object underlying this programme --- the set of +proper $4$-colourings of a boundary cycle that extend to a colouring +of a bounded planar region --- is classical. Birkhoff's reducibility +analysis of the diamond configuration~\cite{birkhoff-reducibility} is +the earliest instance of computing such extension sets to attack the +Four Colour Theorem; the chromatic polynomial framework of Birkhoff +and Lewis~\cite{birkhoff-lewis-chromatic} systematised the counting. +Tutte studied how the chromatic polynomial of a rooted planar +triangulation decomposes along its outer +boundary~\cite{tutte-chromatic-sums-1973} and developed an algebraic +theory of graph colourings organised around separating +subgraphs~\cite{tutte-algebraic-colorings, tutte-four-colour-conjecture}. +The most recent and structurally closest parallel is Dvo\v{r}\'ak +and Lidick\'y's analysis of \emph{coloring count +cones}~\cite{dvorak-lidicky-cones}, which characterises the possible +boundary-extension functions on a fixed outer cycle of a +near-triangulation. The Heesch--Appel--Haken +approach~\cite{heesch-untersuchungen, robertson-sanders-seymour-thomas} +also uses boundary-extension reasoning, but case-by-case on a finite +unavoidable set of local configurations rather than as part of a +global structural induction. + +The tire-tree decomposition used here differs from each of +these in shape rather than ingredients. Birkhoff, Tutte, and +Dvo\v{r}\'ak--Lidick\'y all study \emph{one} boundary; Heesch and +the cleaned-up Appel--Haken proof~\cite{robertson-sanders-seymour-thomas} +study a finite collection of local boundaries. The present framework +organises the entire triangulation into a hierarchy of annular +regions glued along level cycles, and asks whether boundary-extension +constraints compose compatibly up the hierarchy. To the authors' +knowledge, no prior work on the Four Colour Theorem has been +organised around a global nested-cycle decomposition of this kind. + +\section{Background from nested tire decompositions} + +We use the terminology and structural results of~\cite{bauerfeld-nested-tire-decompositions}. In particular, a level source induces levels in a plane maximal planar graph, the depth-$d$ inner-dual components determine tire graphs, and the resulting tire treads form a rooted tire tree $\mathcal{T}(G,S)$. For a tread $T$, we write $B_{\mathrm{out}}^{(T)}$ and $B_{\mathrm{in}}^{(T)}$ for its outer and inner boundary data, $O^{(T)}$ for its inner outerplanar graph, and $G_T$ for the triangulated disk on the descendant side of $B_{\mathrm{out}}^{(T)}$. The base paper also records the boundary-state transfer viewpoint for a single tire and the factorisation of global colouring questions through local tread colourings together with compatibility along parent-child interfaces. + +\begin{remark}[Motivation for level-cycle restrictions] +\label{rem:level-cycle-motivation} +The tire-tree decomposition reduces global colouring questions to local +choices on treads together with compatibility along nested boundary +cycles. Without further structure, the number of boundary colour states +can grow quickly as one descends the tree: each seam or level cycle may +in principle carry any proper restriction of a $4$-colouring. The +following restriction is meant to test whether this state space can be +compressed. If level cycles can always be made to omit one colour, then +each such interface behaves like a three-colour boundary object, while +still allowing different cycles to omit different colours. This would +not by itself solve the gluing problem, but it would give a simpler +target class of boundary states for arguments about nested tire trees. +\end{remark} + +\begin{definition}[Level-cycle three-colour restriction] +\label{def:level-cycle-three-colour-restriction} +Let $G$ be a maximal planar graph, let $S \subseteq V(G)$ be a level +source, and let $c \colon V(G) \to \{1,2,3,4\}$ be a proper +$4$-vertex-colouring of $G$. We say that $c$ has the +\emph{level-cycle three-colour restriction} with respect to $S$ if, +for every level $d \geq 0$ and every simple cycle +$C \subseteq G[L_d]$, the colour set used on $C$ has size at most +three: +\[ + |c(V(C))| \leq 3. +\] +Equivalently, every simple cycle contained in a single level omits at +least one of the four colours. The omitted colour may depend on the +cycle; in particular, distinct cycles in the same level, the same tire +tread, or the same inner outerplanar component are not required to +omit the same colour. +\end{definition} + +\begin{conjecture}[False universal-source form] +\label{conj:false-universal-level-cycle-three-colour} +Let $G$ be a maximal planar graph and let $S \subseteq V(G)$ be any +level source. Then $G$ admits a proper $4$-vertex-colouring with the +level-cycle three-colour restriction with respect to $S$. +\end{conjecture} + +\begin{figure}[htbp] +\centering +\includegraphics[width=0.78\textwidth]{fig_universal_level_cycle_counterexample.png} +\caption{The $8$-vertex counterexample to the universal-source form. +With source $S=\{7\}$, the level cycle $(3,4,5,8)$ lies in $L_2$ and +forces all four colours in every proper $4$-vertex-colouring.} +\label{fig:universal-level-cycle-counterexample} +\end{figure} + +\begin{example}[Counterexample to Conjecture~\ref{conj:false-universal-level-cycle-three-colour}] +\label{ex:universal-level-cycle-counterexample} +Let $G$ be the maximal planar graph on vertex set +$\{1,2,3,4,5,6,7,8\}$ with edge set +\[ +\begin{aligned} +E(G)=\{& +12,13,14,15,16,17,23,26,27,34,35,36,38,\\ +&45,56,58,67,68\}. +\end{aligned} +\] +Here $ij$ denotes the edge $\{i,j\}$. +Take the vertex source $S=\{7\}$. The corresponding levels are +\[ +L_0=\{7\},\qquad L_1=\{1,2,6\},\qquad +L_2=\{3,4,5,8\}. +\] +Inside $G[L_2]$ the vertices $(3,4,5,8)$ form a simple cycle. In +every proper $4$-vertex-colouring of $G$, these four vertices receive +four distinct colours. The edges $34$, $45$, $58$, $38$, and $35$ +force all pairs among $\{3,4,5,8\}$ except possibly $\{4,8\}$ to have +distinct colours. If $4$ and $8$ had the same colour, then vertex $6$, +which is adjacent to $3$, $5$, and $8$, would have to use the fourth +colour; but vertex $1$ is adjacent to $3$, $4$, $5$, and $6$, and +would then be adjacent to all four colours, impossible in a proper +$4$-colouring. Hence $4$ and $8$ also have distinct colours, so the +level cycle $(3,4,5,8)$ uses all four colours in every proper +$4$-colouring of $G$. Therefore no proper $4$-colouring has the +level-cycle three-colour restriction with respect to $S=\{7\}$. +\end{example} + +\subsection*{An inner-boundary refinement} + +The level-cycle restriction constrains \emph{every} simple cycle in +every level. For the tire-tree program, the cycles that actually carry +boundary state are fewer: each tire transfers colour information across +its tread between its two boundaries, so it is the tire +\emph{inner boundaries} $B_{\mathrm{in}}^{(T)}$ --- not all level cycles +--- that one wishes to compress. This motivates a restriction stated +directly in the objects of the decomposition. + +\begin{definition}[Tire inner-boundary three-colour restriction] +\label{def:tire-inner-boundary-three-colour} +Let $G$ be a maximal planar graph, let $v_0 \in V(G)$ be a vertex source +on the outer face of $\Pi_G$, and let $c \colon V(G) \to \{1,2,3,4\}$ be +a proper $4$-vertex-colouring of $G$. We say $c$ has the \emph{tire +inner-boundary three-colour restriction} with respect to +$\mathcal{T}(G, \{v_0\})$ if every tire tread $T \in +\mathcal{T}(G, \{v_0\})$ satisfies +\[ + |c(V(B_{\mathrm{in}}^{(T)}))| \leq 3, +\] +i.e.\ the inner boundary of every tire omits at least one of the four +colours. (A degenerate inner boundary is a single vertex and the +condition is then vacuous.) +\end{definition} + +\begin{conjecture}[Tire inner-boundary three-colour conjecture] +\label{conj:tire-inner-boundary-three-colour} +Every maximal planar graph $G$ admits a vertex source $v_0 \in V(G)$ and +a proper $4$-vertex-colouring $c$ of $G$ such that $c$ has the tire +inner-boundary three-colour restriction with respect to +$\mathcal{T}(G, \{v_0\})$. +\end{conjecture} + +\subsection*{A counterexample at $n=14$} + +Conjecture~\ref{conj:tire-inner-boundary-three-colour} is in fact +false. An exhaustive search over the triangulations enumerated by +\texttt{plantri} at $n=14$ encounters a graph $G^\star$ on $14$ vertices +and $36$ edges --- specifically, the graph at index $263993$ in the +\texttt{plantri} enumeration --- for which no vertex source admits any +witness. + +\begin{example}[Counterexample to Conjecture~\ref{conj:tire-inner-boundary-three-colour}] +\label{ex:inner-boundary-counterexample} +Let $G^\star$ be the maximal planar graph with vertex set +$\{1,2,\dots,14\}$ and edge set +\begin{align*} +E(G^\star) = \{ + & 12, 13, 14, 15, 16, 17, 18, \\ + & 23, 24, 26, 28, 29, 2\,10, \\ + & 34, 45, 46, 4\,10, 56, 67, 69, 6\,10, \\ + & 78, 79, 7\,11, 7\,12, 7\,13, \\ + & 89, 8\,12, 8\,13, 8\,14, \\ + & 9\,11, 9\,12, 9\,14, \\ + & 11\,12, 12\,13, 12\,14 +\}. +\end{align*} +The graph $G^\star$ is a $3$-connected (but not $5$-connected) planar +triangulation with degree sequence +$(7,7,7,7,7,7,6,6,3,3,3,3,3,3)$ and exactly $96$ proper $4$-vertex +colourings. For \emph{every} choice of vertex source +$v_0 \in V(G^\star)$, each of the $96$ proper $4$-colourings of +$G^\star$ has some tire whose inner boundary uses all four colours. +A planar embedding is shown in +Figure~\ref{fig:inner-boundary-counterexample}. +\end{example} + +\begin{figure}[ht] +\centering +\includegraphics[width=0.78\textwidth]{fig_inner_boundary_counterexample} +\caption{The $14$-vertex counterexample $G^\star$ to +Conjecture~\ref{conj:tire-inner-boundary-three-colour} in a planar +embedding. The six degree-$3$ vertices split into two triples, +$\{3,5,10\}$ each adjacent to a triangle in the +core $\{1,2,4,6\}$, and $\{11,13,14\}$ each adjacent to a triangle in +the core $\{7,8,9,12\}$; the two cores are joined by the edges +$17,28,69$ together with $12$.} +\label{fig:inner-boundary-counterexample} +\end{figure} + +The failure was verified by enumerating, for each of the $14$ vertex +sources, all $96$ proper $4$-colourings of $G^\star$ and computing the +inner boundary $V(B_{\mathrm{in}}^{(T)})$ of every tire $T$ as the +level-$(d+1)$ vertices of the corresponding depth-$d$ dual component. +Each source has exactly two non-degenerate inner boundaries +(size $\geq 4$), and every proper $4$-colouring assigns all four +colours to at least one of them. + +The graph $G^\star$ does not refute +Conjecture~\ref{conj:level-cycle-three-colour}: the vertex source +$v_0 = 10$ admits a proper $4$-colouring under which every simple level +cycle uses at most three colours. + +\subsection*{The surviving level-cycle conjecture} + +\begin{conjecture}[Level-cycle three-colour conjecture] +\label{conj:level-cycle-three-colour} +Let $G$ be a maximal planar graph. Then there exists a level source +$S \subseteq V(G)$ such that $G$ admits a proper $4$-vertex-colouring +with the level-cycle three-colour restriction with respect to $S$. +\end{conjecture} + +\subsection*{Enumeration for small $n$} + +We exhaustively enumerated all plane triangulation isomorphism classes with +$4 \leq n \leq 13$ vertices and searched the vertex sources for each graph. +No counterexample to Conjecture~\ref{conj:level-cycle-three-colour} appeared +in this range. Table~\ref{tab:level-cycle-three-colour-counts} records the +size of the search space and the number of triangulations that admit a +witness. + +\begin{table}[ht] +\centering +\small +\setlength{\tabcolsep}{4pt} +\begin{tabular}{ccc} +$n$ & triangulations & with witness \\\hline +$4$ & $1$ & $1$ \\ +$5$ & $1$ & $1$ \\ +$6$ & $2$ & $2$ \\ +$7$ & $5$ & $5$ \\ +$8$ & $14$ & $14$ \\ +$9$ & $50$ & $50$ \\ +$10$ & $233$ & $233$ \\ +$11$ & $1249$ & $1249$ \\ +$12$ & $7595$ & $7595$ \\ +$13$ & $49566$ & $49566$ \\ +\end{tabular} +\caption{Exhaustive vertex-source search for the level-cycle three-colour conjecture on all triangulation isomorphism classes with $4 \leq n \leq 13$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.} +\label{tab:level-cycle-three-colour-counts} +\end{table} + +We also tested the six dual triangulations of the Holton--McKay graphs, +which lie just beyond this census, and found witnesses in each case. + +\subsection*{The $5$-connected slice at $n \leq 24$} + +As a compact test above the full small-$n$ census, we also enumerated the +$5$-connected triangulations at $14 \leq n \leq 24$ with \texttt{plantri +-c5 -a}. These are especially rigid triangulations, and the slice remains +small enough to check exhaustively. Every graph in this slice admits a +vertex source witnessing Conjecture~\ref{conj:level-cycle-three-colour}. + +\begin{table}[ht] +\centering +\small +\setlength{\tabcolsep}{4pt} +\begin{tabular}{ccc} +$n$ & $5$-connected triangulations & with witness \\\hline +$14$ & $1$ & $1$ \\ +$15$ & $1$ & $1$ \\ +$16$ & $3$ & $3$ \\ +$17$ & $4$ & $4$ \\ +$18$ & $12$ & $12$ \\ +$19$ & $23$ & $23$ \\ +$20$ & $71$ & $71$ \\ +$21$ & $187$ & $187$ \\ +$22$ & $627$ & $627$ \\ +$23$ & $1970$ & $1970$ \\ +$24$ & $6833$ & $6833$ \\ +\end{tabular} +\caption{The $5$-connected triangulations at $14 \leq n \leq 24$ generated by +\texttt{plantri -c5 -a}. All $9732$ graphs in this slice admit a vertex +source witnessing the level-cycle three-colour conjecture.} +\label{tab:level-cycle-three-colour-c5-14-16} +\end{table} + +\begin{definition}[Seam] +\label{def:seam} +A \emph{seam} of a maximal planar graph $G$ is a simple cycle +$C \subset G$ such that, for some vertex $v_0 \in V(G)$, $C = +B_{\mathrm{out}}^{(T)}$ for some non-root tread $T$ in +$\mathcal{T}(G, \{v_0\})$. + +By the tire-tree decomposition theorem of +\cite{bauerfeld-nested-tire-decompositions}, every seam $C$ separates +$G$ into: +\begin{itemize} +\item the \emph{seam interior} $G_T$, the triangulated disk on the + $T$-descendant side of $C$; +\item the \emph{seam exterior} $G_C^{\mathrm{ext}} := G \setminus + \mathrm{int}(G_T)$, the triangulated polygon with outer face + bounded by $C$ on the side containing $v_0$; +\end{itemize} +both sharing $C$. A seam is \emph{non-trivial} if both +$V(G_T) \setminus V(C)$ and $V(G_C^{\mathrm{ext}}) \setminus V(C)$ are +non-empty. + +For any seam $C$ and either side $X \in \{G_T, G_C^{\mathrm{ext}}\}$, +write +\[ + \mathrm{Col}(X \mid C) \;:=\; \bigl\{\, c|_{V(C)} \;:\; c \text{ a + proper $4$-colouring of } X \,\bigr\} \;\subseteq\; \{1,2,3,4\}^{V(C)} +\] +for the set of $C$-restricted $4$-colourings induced by $4$-colourings +of $X$ (each element is a proper $4$-colouring of the cycle $C$). +\end{definition} + +\begin{definition}[Partial tire tree] +\label{def:partial-tire-tree} +Let $T_r$ be a tire tread in $\mathcal{T}(G, S)$ with outer boundary +cycle $C_{T_r} = B_{\mathrm{out}}^{(T_r)}$, and let $G_{T_r}$ be the +triangulated disk inside $C_{T_r}$ given by the tire-tree +decomposition theorem of~\cite{bauerfeld-nested-tire-decompositions}. +The \emph{partial tire +tree} with root $T_r$, written $G_{T_r}^{\circ}$, is the induced +subgraph of $G$ on the vertex set +$V(G_{T_r}) \setminus V(C_{T_r})$ --- +i.e.\ $G_{T_r}$ with the seam-cycle vertices removed. + +Equivalently, $V(G_{T_r}^{\circ})$ is the set of vertices of $G$ +strictly inside $C_{T_r}$ on the side away from the level source, +and $E(G_{T_r}^{\circ})$ consists of the edges of $G$ both of whose +endpoints lie in this strict interior. The tree-of-tire-treads +structure of $G_{T_r}^{\circ}$ is the sub-tree of $\mathcal{T}(G, S)$ +rooted at $T_r$, with $T_r$'s outer boundary peeled away. +\end{definition} + +\begin{lemma}[Seam edges are shared by at most one other depth-$d$ seam] +\label{lem:seam-edge-shared} +Let $G$ be a maximal planar graph with single-vertex level source +$S = \{v_0\}$, fix $d \ge 1$, and let $e \in E(G)$ be an edge lying on +the seam $C_T = B_{\mathrm{out}}^{(T)}$ of some tire tread +$T \in \mathcal{T}(G, S)$ at depth $d$. Then there is at most one +other tire tread $T' \in \mathcal{T}(G, S)$ at the same depth $d$ with +$e \in C_{T'}$. +\end{lemma} + +\begin{proof} +By the child--face correspondence in the tire-tree construction of +\cite{bauerfeld-nested-tire-decompositions}, $C_T$ is the boundary cycle of a +bounded face of the parent's inner outerplanar graph $O^{(T_p)}$, +where $T_p \in \mathcal{T}(G, S)$ is the parent of $T$ at depth +$d - 1$. The inner dual of a connected outerplanar graph is a tree, +so each edge of +$O^{(T_p)}$ lies on at most two of its bounded face cycles. Hence +$e$ lies on at most one other bounded face cycle of $O^{(T_p)}$, +corresponding, by the same child--face bijection, +to at most one sibling of $T$ at depth $d$ whose seam contains $e$. +\end{proof} + +\begin{conjecture}[Seam structure of minimum $4$CT counterexamples, sketch] +\label{conj:seam-counterexample} +Suppose the Four Colour Theorem fails: there exists a maximal planar +graph that is not $4$-colourable. Let $G$ be a \emph{minimum} such +counterexample (with $|V(G)|$ minimal among non-$4$-colourable maximal +planar graphs). Then: + +\medskip + +\noindent\emph{Restatement-of-classical content.} +\begin{itemize} +\item[(C1)] \emph{Bilateral colourability.} For every non-trivial seam + $C$ of $G$, both $\mathrm{Col}(G_T \mid C)$ and + $\mathrm{Col}(G_C^{\mathrm{ext}} \mid C)$ are non-empty. +\item[(C2)] \emph{Bilateral incompatibility.} For every non-trivial + seam $C$, + \[ + \mathrm{Col}(G_T \mid C) \;\cap\; + \mathrm{Col}(G_C^{\mathrm{ext}} \mid C) \;=\; \emptyset. + \] +\item[(C3)] \emph{Length lower bound (Birkhoff).} Every non-trivial + seam $C$ of $G$ has $|V(C)| \ge 6$. +\end{itemize} + +(C1) and (C2) together restate ``$G$ is a counterexample whose every +internal cut by a seam splits into two colourable pieces with +incompatible boundary palettes''; (C1) follows from minimality applied +to each side after closing the polygonal outer face by a single apex, +(C2) from $G$ itself being non-$4$-colourable. (C3) is Birkhoff's +internally-$6$-connected condition restated in the seam language. + +\medskip + +\noindent\emph{Substantive (speculative) content.} +\begin{itemize} +\item[(C4)] \emph{Innermost obstruction.} There exists a vertex source + $v_0 \in V(G)$ and a \emph{leaf} tread $T^* \in + \mathcal{T}(G, \{v_0\})$ (a tread with no children in the + tree-of-treads) such that: + \begin{enumerate} + \item[(i)] the seam interior $G_{T^*}$ is, up to plane + iso, one of a finite list of \emph{minimal seam + configurations}, characterized by their boundary + palette $\mathrm{Col}(G_{T^*} \mid C_{T^*})$ being a + specific proper subset of the proper $4$-colourings + of the cycle $C_{T^*}$; + \item[(ii)] the path in $\mathcal{T}(G, \{v_0\})$ from the + root $T_0$ to $T^*$ is an \emph{obstruction chain}: + $\mathrm{Col}(G_T \mid C_T)$ is monotonically + restricted (under the natural pull-back along + parent--child seams of + the parent--child seam pull-back described in + \cite{bauerfeld-nested-tire-decompositions}) as $T$ + descends from the root to $T^*$, with the final + restriction at $T^*$ being incompatible with the + $v_0$-side palette. + \end{enumerate} +\end{itemize} +\end{conjecture} + +\begin{thebibliography}{9} + +\bibitem{tait-original} +P.~G.~Tait, +\emph{Remarks on the colouring of maps}, +Proc.\ Roy.\ Soc.\ Edinburgh \textbf{10} (1880), 729. + +\bibitem{bauerfeld-depth} +E.~Bauerfeld, +\emph{Plane Depth}, +manuscript (math-research repository), 2026. + +\bibitem{bauerfeld-nested-tire-decompositions} +E.~Bauerfeld, +\emph{Nested Tire Decompositions of Plane Triangulations}, +manuscript (math-research repository), 2026. + +\bibitem{bauerfeld-nested-tire-duals} +E.~Bauerfeld, +\emph{Coloring Nested Tire Dual Graphs}, +manuscript (math-research repository), 2026. + +\bibitem{birkhoff-reducibility} +G.~D.~Birkhoff, +\emph{The reducibility of maps}, +Amer.\ J.\ Math.\ \textbf{35} (1913), 115--128. + +\bibitem{birkhoff-lewis-chromatic} +G.~D.~Birkhoff and D.~C.~Lewis, +\emph{Chromatic polynomials}, +Trans.\ Amer.\ Math.\ Soc.\ \textbf{60} (1946), 355--451. + +\bibitem{tutte-four-colour-conjecture} +W.~T.~Tutte, +\emph{On the four-colour conjecture}, +Proc.\ London Math.\ Soc.\ (2) \textbf{50} (1948), 137--149. + +\bibitem{tutte-algebraic-colorings} +W.~T.~Tutte, +\emph{On the algebraic theory of graph colorings}, +J.\ Combin.\ Theory \textbf{1} (1966), 15--50. + +\bibitem{tutte-chromatic-sums-1973} +W.~T.~Tutte, +\emph{Chromatic sums for rooted planar triangulations: the cases $\lambda = 1$ and $\lambda = 2$}, +Canad.\ J.\ Math.\ \textbf{25} (1973), 426--447. + +\bibitem{heesch-untersuchungen} +H.~Heesch, +\emph{Untersuchungen zum Vierfarbenproblem}, +Hochschulskriptum 810/a/b, Bibliographisches Institut, Mannheim, 1969. + +\bibitem{robertson-sanders-seymour-thomas} +N.~Robertson, D.~P.~Sanders, P.~D.~Seymour, and R.~Thomas, +\emph{The four-colour theorem}, +J.\ Combin.\ Theory Ser.\ B \textbf{70} (1997), 2--44. + +\bibitem{dvorak-lidicky-cones} +Z.~Dvo\v{r}\'ak and B.~Lidick\'y, +\emph{Coloring count cones of planar graphs}, +J.\ Graph Theory \textbf{100} (2022), 84--100. + +\end{thebibliography} + +\end{document}