diff --git a/README.md b/README.md index dd3750c..f2733c7 100644 --- a/README.md +++ b/README.md @@ -11,17 +11,17 @@ All papers are at `papers//paper.tex`. The current set: | `colored_edge_flip_classes` | Colored Edge Flip Classes | | `colored_pentagon_contractions` | Colored Pentagon Reductions | | `coloring_nested_tire_dual_graphs` | Coloring Nested Tire Dual Graphs | -| `coloring_nested_tire_graphs` | Coloring Nested Tire Graphs | | `even_level_graph_generators` | Even Level Graph Generators: a constructive conjecture stronger than the Four Color Theorem | | `face_monochromatic_pairs` | Face-Monochromatic Pairs and the Four Colour Theorem | | `iterated_reduction_in_reduced_dual` | An Iterated Reduction in the Reduced Dual | | `level_resolutions_of_maximal_planar_graphs` | Level Resolutions of Maximal Planar Graphs | | `level_switching` | Level Switching | +| `nested_tire_decompositions_of_plane_triangulations` | Nested Tire Decompositions of Plane Triangulations | | `plane_depth` | Plane Depth | | `plane_depth_sequencing` | Plane Depth Sequencing | | `plane_diamond_coloring` | Plane Diamond Coloring | -The papers form a connected programme around plane triangulations, BFS-level structure, and the Four Colour Theorem. `plane_depth` introduces the level / dual-depth framework that downstream papers build on; `coloring_nested_tire_graphs` develops the tire-tread tree-of-treads decomposition. +The papers form a connected programme around plane triangulations, BFS-level structure, and the Four Colour Theorem. `plane_depth` introduces the level / dual-depth framework that downstream papers build on; `nested_tire_decompositions_of_plane_triangulations` develops the tire-tread tree-of-treads decomposition. ## Creating a New Paper diff --git a/papers/coloring_nested_tire_dual_graphs/paper.fdb_latexmk b/papers/coloring_nested_tire_dual_graphs/paper.fdb_latexmk index 0923669..359a05f 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.fdb_latexmk +++ b/papers/coloring_nested_tire_dual_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1779735598 "paper.tex" "paper.pdf" "paper" 1779735599 +["pdflatex"] 1780945675 "paper.tex" "paper.pdf" "paper" 1780945676 "/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 "" @@ -19,20 +19,29 @@ "/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/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 "" 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/usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt @@ -225,30 +225,44 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7 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 -INPUT ./fig_dual_depth.png -INPUT ./fig_dual_depth.png -INPUT fig_dual_depth.png -INPUT ./fig_dual_depth.png OUTPUT paper.pdf -INPUT ./fig_dual_depth.png INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map -INPUT ./fig_tire_example.png -INPUT ./fig_tire_example.png -INPUT fig_tire_example.png -INPUT ./fig_tire_example.png -INPUT ./fig_tire_example.png +INPUT ./fig_partial_tire_dual.png +INPUT ./fig_partial_tire_dual.png +INPUT fig_partial_tire_dual.png +INPUT ./fig_partial_tire_dual.png +INPUT ./fig_partial_tire_dual.png +INPUT ./fig_partial_tire_dual_bridge.png +INPUT ./fig_partial_tire_dual_bridge.png +INPUT fig_partial_tire_dual_bridge.png +INPUT ./fig_partial_tire_dual_bridge.png +INPUT ./fig_partial_tire_dual_bridge.png +INPUT ./notes/fig_facial_dual_choices.png +INPUT ./notes/fig_facial_dual_choices.png +INPUT notes/fig_facial_dual_choices.png +INPUT ./notes/fig_facial_dual_choices.png +INPUT ./notes/fig_facial_dual_choices.png +INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.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/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/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/cmr10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.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/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/cmsy6.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/cmsy8.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/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index a3e5946..f82cf29 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.log +++ b/papers/coloring_nested_tire_dual_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) 27 MAY 2026 01:13 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 8 JUN 2026 15:07 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -265,7 +265,7 @@ public/amsfonts/cm/cmti10.pfb> -Output written on paper.pdf (7 pages, 583446 bytes). +Output written on paper.pdf (7 pages, 583433 bytes). PDF statistics: 145 PDF objects out of 1000 (max. 8388607) 85 compressed objects within 1 object stream diff --git a/papers/coloring_nested_tire_dual_graphs/paper.pdf b/papers/coloring_nested_tire_dual_graphs/paper.pdf index eb16180..72e27a6 100644 Binary files a/papers/coloring_nested_tire_dual_graphs/paper.pdf and b/papers/coloring_nested_tire_dual_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_dual_graphs/paper.tex b/papers/coloring_nested_tire_dual_graphs/paper.tex index 5264366..6ecdd04 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.tex +++ b/papers/coloring_nested_tire_dual_graphs/paper.tex @@ -469,7 +469,7 @@ manuscript (math-research repository), 2026. \bibitem{bauerfeld-nested-tires} E.~Bauerfeld, -\emph{Coloring Nested Tire Graphs}, +\emph{Nested Tire Decompositions of Plane Triangulations}, manuscript (math-research repository), 2026. \end{thebibliography} diff --git a/papers/coloring_nested_tire_graphs/experiments/check_inner_boundary_on_holton_mckay.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/check_inner_boundary_on_holton_mckay.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/check_inner_boundary_on_holton_mckay.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/check_inner_boundary_on_holton_mckay.py diff --git a/papers/coloring_nested_tire_graphs/experiments/check_level_cycle_three_color.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/check_level_cycle_three_color.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/check_level_cycle_three_color.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/check_level_cycle_three_color.py diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_inner_boundary_counterexample.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_inner_boundary_counterexample.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/draw_inner_boundary_counterexample.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_inner_boundary_counterexample.py diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_medial_tire_graph.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_medial_tire_graph.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/draw_medial_tire_graph.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_medial_tire_graph.py diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_seam_construction.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_seam_construction.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/draw_seam_construction.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_seam_construction.py diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_tire_tree_decomposition.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_tire_tree_decomposition.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/draw_tire_tree_decomposition.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_tire_tree_decomposition.py diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_universal_level_cycle_counterexample.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_universal_level_cycle_counterexample.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/draw_universal_level_cycle_counterexample.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/draw_universal_level_cycle_counterexample.py diff --git a/papers/coloring_nested_tire_graphs/experiments/generate_candidate_families.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/generate_candidate_families.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/generate_candidate_families.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/generate_candidate_families.py diff --git a/papers/coloring_nested_tire_graphs/experiments/kempe_classes.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/kempe_classes.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/kempe_classes.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/kempe_classes.py diff --git a/papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py similarity index 100% rename from papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py diff --git a/papers/coloring_nested_tire_graphs/experiments/level_cycle_support_findings.md b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support_findings.md similarity index 95% rename from papers/coloring_nested_tire_graphs/experiments/level_cycle_support_findings.md rename to papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support_findings.md index 09410c6..0f8b43d 100644 --- a/papers/coloring_nested_tire_graphs/experiments/level_cycle_support_findings.md +++ b/papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support_findings.md @@ -10,7 +10,7 @@ the outer-face boundary of `O`. Executable: ```bash -python3 papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py +python3 papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py ``` ## Model @@ -35,7 +35,7 @@ of `C`. Command: ```bash -python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \ +python3 -u papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py \ --n-min 3 --n-max 6 --outer-max 5 --inner-max 7 --max-chords 2 --max-paths 30 ``` @@ -133,7 +133,7 @@ window for `n = 6`. The smallest admissible inner ring is `k = 7` Command: ```bash -python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \ +python3 -u papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py \ --n-min 6 --n-max 6 --outer-min 3 --outer-max 6 --inner-min 7 --inner-max 8 \ --progress --examples 6 ``` @@ -175,7 +175,7 @@ strictly *less* restrictive supports — never a new floor. Command: ```bash -python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \ +python3 -u papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py \ --n-min 6 --n-max 6 --outer-min 3 --outer-max 7 --inner-min 7 --inner-max 9 \ --progress --examples 6 ``` @@ -246,7 +246,7 @@ To probe the conjecture more directly we ran a "skip-`m=3`" sweep explicitly excluding `m = 3` from the search: ```bash -python3 -u papers/coloring_nested_tire_graphs/experiments/level_cycle_support.py \ +python3 -u papers/nested_tire_decompositions_of_plane_triangulations/experiments/level_cycle_support.py \ --n-min 6 --n-max 6 --outer-min 4 --outer-max 10 --inner-min 7 --inner-max 9 \ --progress ``` diff --git a/papers/coloring_nested_tire_graphs/fig_dual_depth.png b/papers/nested_tire_decompositions_of_plane_triangulations/fig_dual_depth.png similarity index 100% rename from papers/coloring_nested_tire_graphs/fig_dual_depth.png rename to papers/nested_tire_decompositions_of_plane_triangulations/fig_dual_depth.png diff --git a/papers/coloring_nested_tire_graphs/fig_inner_boundary_counterexample.png b/papers/nested_tire_decompositions_of_plane_triangulations/fig_inner_boundary_counterexample.png similarity index 100% rename from papers/coloring_nested_tire_graphs/fig_inner_boundary_counterexample.png rename to papers/nested_tire_decompositions_of_plane_triangulations/fig_inner_boundary_counterexample.png diff --git a/papers/coloring_nested_tire_graphs/fig_medial_tire_example.png b/papers/nested_tire_decompositions_of_plane_triangulations/fig_medial_tire_example.png similarity index 100% rename from papers/coloring_nested_tire_graphs/fig_medial_tire_example.png rename to papers/nested_tire_decompositions_of_plane_triangulations/fig_medial_tire_example.png diff --git a/papers/coloring_nested_tire_graphs/fig_seam_construction.png b/papers/nested_tire_decompositions_of_plane_triangulations/fig_seam_construction.png similarity index 100% rename from papers/coloring_nested_tire_graphs/fig_seam_construction.png rename to papers/nested_tire_decompositions_of_plane_triangulations/fig_seam_construction.png diff --git a/papers/coloring_nested_tire_graphs/fig_tire_example.png b/papers/nested_tire_decompositions_of_plane_triangulations/fig_tire_example.png similarity index 100% rename from papers/coloring_nested_tire_graphs/fig_tire_example.png rename to papers/nested_tire_decompositions_of_plane_triangulations/fig_tire_example.png diff --git a/papers/coloring_nested_tire_graphs/fig_tire_tree_decomposition.png b/papers/nested_tire_decompositions_of_plane_triangulations/fig_tire_tree_decomposition.png similarity index 100% rename from papers/coloring_nested_tire_graphs/fig_tire_tree_decomposition.png rename to papers/nested_tire_decompositions_of_plane_triangulations/fig_tire_tree_decomposition.png diff --git a/papers/coloring_nested_tire_graphs/fig_universal_level_cycle_counterexample.png b/papers/nested_tire_decompositions_of_plane_triangulations/fig_universal_level_cycle_counterexample.png similarity index 100% rename from papers/coloring_nested_tire_graphs/fig_universal_level_cycle_counterexample.png rename to papers/nested_tire_decompositions_of_plane_triangulations/fig_universal_level_cycle_counterexample.png diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/nested_tire_decompositions_of_plane_triangulations/paper.aux similarity index 90% rename from papers/coloring_nested_tire_graphs/paper.aux rename to papers/nested_tire_decompositions_of_plane_triangulations/paper.aux index f8960e4..9caf7f0 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/nested_tire_decompositions_of_plane_triangulations/paper.aux @@ -40,15 +40,12 @@ \newlabel{fig:inner-dual-annulus-case}{{5}{11}} \newlabel{rem:hamilton-cycle-spoke-only}{{1.16}{11}} \newlabel{rem:bridge-case-theta}{{1.17}{11}} -\citation{tait-original} -\newlabel{thm:tait-tire}{{1.18}{12}} -\newlabel{rem:count-general-outerplanar}{{1.19}{12}} -\newlabel{def:boundary-state-transfer}{{1.20}{13}} -\newlabel{thm:tire-chromatic-polynomial-transfer}{{1.21}{13}} -\newlabel{rem:spoke-only-chromatic-transfer}{{1.22}{14}} -\newlabel{thm:tread-tree}{{1.23}{14}} -\newlabel{rem:tree-multiple-children}{{1.24}{15}} -\newlabel{thm:tire-tree-decomposition}{{1.25}{15}} +\newlabel{thm:tread-tree}{{1.18}{12}} +\newlabel{rem:tree-multiple-children}{{1.19}{13}} +\newlabel{thm:tire-tree-decomposition}{{1.20}{13}} +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.20\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 $(a)$.}}{15}{}\protected@file@percent } +\newlabel{fig:tire-tree-decomposition}{{6}{15}} +\newlabel{rem:tree-coloring-factorisation}{{1.21}{15}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} @@ -65,8 +62,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\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} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{16}{}\protected@file@percent } +\gdef \@abspage@last{16} diff --git a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk b/papers/nested_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk similarity index 98% rename from papers/coloring_nested_tire_graphs/paper.fdb_latexmk rename to papers/nested_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk index a47c09c..d9afbbb 100644 --- a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk +++ b/papers/nested_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1780944443 "paper.tex" "paper.pdf" "paper" 1780944444 +["pdflatex"] 1780945550 "paper.tex" "paper.pdf" "paper" 1780945552 "/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 "" @@ -143,8 +143,8 @@ "fig_medial_tire_example.png" 1780943640 209003 4349824e4f016bde4b938be6b2cb5b2c "" "fig_tire_example.png" 1779857443 104494 8f9ce26b469b4236b8b67829f73a5faa "" "fig_tire_tree_decomposition.png" 1780290287 372371 1b44f5a3e9f637d78ae951b1f2e3a89d "" - "paper.aux" 1780944444 7237 ad56df69e173b3ef29a32273960fe919 "pdflatex" - "paper.tex" 1780944408 69767 384639d36c3a81504fdd37ae395860bb "" + "paper.aux" 1780945552 6952 b88a9a8b764572f2fc3be94aabc8b157 "pdflatex" + "paper.tex" 1780945491 60644 1d96ad32143c562e73fccdfacbbc9be2 "" (generated) "paper.aux" "paper.log" diff --git a/papers/coloring_nested_tire_graphs/paper.fls b/papers/nested_tire_decompositions_of_plane_triangulations/paper.fls similarity index 99% rename from papers/coloring_nested_tire_graphs/paper.fls rename to papers/nested_tire_decompositions_of_plane_triangulations/paper.fls index 39e0229..615a2ee 100644 --- a/papers/coloring_nested_tire_graphs/paper.fls +++ b/papers/nested_tire_decompositions_of_plane_triangulations/paper.fls @@ -1,4 +1,4 @@ -PWD /Users/didericis/Code/math-research/papers/coloring_nested_tire_graphs +PWD /Users/didericis/Code/math-research/papers/nested_tire_decompositions_of_plane_triangulations 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 diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/nested_tire_decompositions_of_plane_triangulations/paper.log similarity index 89% rename from papers/coloring_nested_tire_graphs/paper.log rename to papers/nested_tire_decompositions_of_plane_triangulations/paper.log index d150ce5..3948fb5 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/nested_tire_decompositions_of_plane_triangulations/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 14:47 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 8 JUN 2026 15:05 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -524,55 +524,55 @@ LaTeX Warning: `h' float specifier changed to `ht'. 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PDF statistics: - 195 PDF objects out of 1000 (max. 8388607) - 117 compressed objects within 2 object streams + 189 PDF objects out of 1000 (max. 8388607) + 113 compressed objects within 2 object streams 0 named destinations out of 1000 (max. 500000) 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/nested_tire_decompositions_of_plane_triangulations/paper.pdf similarity index 86% rename from papers/coloring_nested_tire_graphs/paper.pdf rename to papers/nested_tire_decompositions_of_plane_triangulations/paper.pdf index d36f1f2..bc0918a 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/nested_tire_decompositions_of_plane_triangulations/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/nested_tire_decompositions_of_plane_triangulations/paper.tex similarity index 86% rename from papers/coloring_nested_tire_graphs/paper.tex rename to papers/nested_tire_decompositions_of_plane_triangulations/paper.tex index 9f06101..d84f606 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/nested_tire_decompositions_of_plane_triangulations/paper.tex @@ -894,185 +894,6 @@ and so contributes no degree-$2$ branch vertex), hence is outerplanar as predicted. \end{remark} -\begin{theorem}[Tait correspondence: 4-colorings of a tire vs 3-edge-colorings of its inner dual] -\label{thm:tait-tire} -Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph -(viewed as an annular triangulation of its tire tread $R$) and let -$\Gamma$ be its inner dual -(Theorem~\ref{thm:inner-dual-outerplanar}). Then -\[ - \#\bigl\{\text{proper $4$-vertex-colorings of } T\bigr\} \big/ |S_4| - \;=\; - \#\bigl\{\text{proper $3$-edge-colorings of } \Gamma\bigr\} \big/ |S_3|. -\] -That is, the number of $4$-vertex-colorings of $T$ up to permutation -of the colour set $\{0, 1, 2, 3\}$ equals the number of -$3$-edge-colorings of $\Gamma$ up to permutation of the colour set -$\{1, 2, 3\}$. -\end{theorem} - -\begin{proof} -The argument is the classical Tait correspondence -\cite{tait-original} adapted to the annular triangulation $T$. -Encode the four colours of a proper $4$-vertex-coloring $c \colon -V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$. For each interior -annular edge $e$ of $T$ (whose two incident faces both lie in -$F_{\mathrm{ann}}$, contributing a $\Gamma$-edge $e^*$), set -\[ - \chi^*(e^*) \;:=\; c(u) + c(v) \in \mathbb{Z}_2 \times \mathbb{Z}_2, - \qquad \text{where } u, v \text{ are the endpoints of } e. -\] -Since $c(u) \ne c(v)$, we have $\chi^*(e^*) \ne 00$, so $\chi^*$ -takes values in $\{01, 10, 11\}$, which we identify with the -$3$-edge-coloring palette $\{1, 2, 3\}$. - -\emph{Properness.} At each $\Gamma$-vertex $d_f$ corresponding to -an annular triangle $f = \{u, v, w\}$, the three incident -$\Gamma$-edges (one per cycle-edge of $f$) carry colours -$c(u) + c(v),\; c(v) + c(w),\; c(u) + c(w)$. These three elements -of $\mathbb{Z}_2 \times \mathbb{Z}_2$ sum to $0$ and are pairwise -distinct (their pairwise differences are -$c(u) - c(w),\; c(v) - c(u),\; c(w) - c(v)$, each nonzero), so -they form a permutation of $\{01, 10, 11\}$ --- a proper edge -colouring at $d_f$. - -\emph{Surjectivity onto cosets.} Given a proper $3$-edge-coloring -$\chi^*$ of $\Gamma$, the equation $c(u) + c(v) = \chi^*(e^*)$ -admits exactly $|\mathbb{Z}_2 \times \mathbb{Z}_2| = 4$ solutions -$c \colon V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$ (a global -translation is the only freedom). Hence the map $c \mapsto -\chi^*$ is $4$-to-$1$. - -\emph{Count.} Therefore $\#\{\text{4-colorings of } T\} = 4 \cdot -\#\{\text{3-edge-colorings of } \Gamma\}$. Dividing by $|S_4| -= 24$ on the left and $|S_3| = 6$ on the right (since $S_4$ acts -faithfully on the $4$-colorings and $S_3$ on the $3$-edge-colorings, -and the $4$-to-$1$ map respects the $S_4/S_3 \cong S_3$ quotient -via the natural surjection $S_4 \twoheadrightarrow S_3$) gives the -stated equality. -\end{proof} - -\begin{remark} -\label{rem:count-general-outerplanar} -Theorem~\ref{thm:tait-tire} reduces the $4$-colouring count of a -tire to the $3$-edge-coloring count of its outerplanar inner dual -$\Gamma$. For the cycle case $\Gamma \cong C_{\mu+\nu}$ (the spoke-only -case of Remark~\ref{rem:hamilton-cycle-spoke-only}), the cycle -chromatic polynomial at $3$ colours gives -$2^{\mu+\nu} + 2 (-1)^{\mu+\nu}$. For an inner dual with one or more -non-crossing chords, the count depends on the chord structure, not just -on the pair (number of vertices, number of chords): two outerplanar -graphs with the same number of vertices and number of chords can have -different proper -$3$-edge-coloring counts depending on how the chords are arranged -(nested, sequential, sharing vertices, etc.). Every such count -can nevertheless be computed in linear time by tree-decomposition -methods, since outerplanar graphs have treewidth at most $2$ and -the edge-chromatic polynomial admits a deletion--contraction -recursion that respects the cycle-plus-chord structure. -\end{remark} - -\begin{definition}[Boundary-state chromatic transfer] -\label{def:boundary-state-transfer} -Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph. -Choose a cut along one annular edge if both boundaries are -non-degenerate; in the degenerate case make no cut. The tread becomes -a triangulated disk $\widetilde R$. Let -\[ - f_1, f_2, \ldots, f_m -\] -be any shelling order of the triangular faces of $\widetilde R$, i.e.\ -an order in which each initial union -$\widetilde R_i := f_1 \cup \cdots \cup f_i$ is a disk. Such an order -is obtained by taking an outerplanar embedding of the inner dual -$\Gamma$ from Theorem~\ref{thm:inner-dual-outerplanar} and repeatedly -removing an outer-face ear. - -For each $i$, let $A_i$ be the \emph{frontier}: the vertices of -$T$ incident to at least one processed face in $\widetilde R_i$ and to -at least one still-unprocessed constraint, where the unprocessed -constraints are the remaining annular faces together with any edge of -$O$ not yet tested by the transfer. A \emph{boundary state} on $A_i$ -is a partition $\pi$ of $A_i$ into colour classes, subject to the -condition that adjacent vertices of $T[A_i]$ lie in distinct blocks. -We write $r(\pi)$ for the number of blocks of $\pi$. -\end{definition} - -\begin{theorem}[Chromatic polynomial of a tire by frontier transfer] -\label{thm:tire-chromatic-polynomial-transfer} -For every tire graph $T$, the chromatic polynomial $P_T(q)$ is -computed by the following boundary-state dynamic program. - -Initialize the table at $i=0$ with the empty frontier state of weight -$1$. When the next triangular face -$f_i = \{x,y,z\}$ is attached, pass from states on $A_{i-1}$ to states -on $A_i$ as follows. -\begin{enumerate} -\item[(1)] Introduce any vertices of $f_i$ not already present in - $A_{i-1}$, assigning each such vertex either to an existing - colour block not containing one of its already-coloured - neighbours, or to a new block. -\item[(2)] Reject every assignment in which two adjacent vertices of - the triangle $f_i$ lie in the same block. Also reject every - assignment in which an edge of $O$ whose two endpoints have - now both appeared for the first time as a tested pair has - both endpoints in the same block. Thus chords and bridges of - the inner outerplanar graph are enforced exactly when their - second endpoint becomes visible to the transfer. -\item[(3)] Delete from the state every vertex no longer incident to an - unprocessed constraint. If deleting a vertex removes the last - representative of its colour block from the frontier, multiply - that transition by $1$; the colour has already been chosen. -\item[(4)] If a new vertex is assigned to a new colour block while the - current frontier state has $r$ colour blocks, multiply that - transition by $q-r$. If several new colour blocks are created - in the same triangle, the factors are - $(q-r)(q-r-1)\cdots$ in the order of creation. -\end{enumerate} -After $f_m$ is processed, the frontier is empty. The single remaining -weight is $P_T(q)$. -\end{theorem} - -\begin{proof} -The construction is the standard transfer for the chromatic polynomial, -specialized to the tire shelling. The frontier state records exactly -the equality pattern among colours that can still affect unprocessed -faces. Since colour names are irrelevant to the chromatic polynomial, -states are quotiented by the natural action of the symmetric group on -the colour set; a state with $r$ visible colour blocks can be extended -by a genuinely new colour in $q-r$ ways. - -Each transition accounts for all proper colourings of the enlarged -processed disk $\widetilde R_i$ that restrict to the resulting frontier -state, and accounts for none that violate an edge of the newly attached -triangle or untested edge of $O$. Vertices removed from the frontier -have no future incident unprocessed constraint, so their actual colour -names can no longer influence compatibility and may be forgotten. -Induction on $i$ therefore shows -that the table after step $i$ is precisely the orbit-count generating -function for proper colourings of $\widetilde R_i$ by frontier state. -At $i=m$ no vertices remain active, so the accumulated weight counts all -proper colourings of $T$. Because the weights are polynomials in $q$, -this count is the full chromatic polynomial. -\end{proof} - -\begin{remark}[Spoke-only transfer matrix] -\label{rem:spoke-only-chromatic-transfer} -In the spoke-only case with both boundaries simple cycles, the method -has a particularly small form. Cut the annulus along one spoke and -walk around the resulting strip. Each step adds one triangle sharing -an edge with the previous processed strip, so the frontier consists of -two or three consecutive boundary vertices. Up to colour permutation -there are only the possible equality patterns among those active -vertices, with adjacent vertices required to be distinct. The -chromatic polynomial is therefore the trace of a finite transfer matrix -whose entries are polynomials in $q$; the matrix depends only on the -local triangle type encountered while walking around the tread. Chords -or cut-vertices of $O$ enlarge the frontier only at the corresponding -outerplanar ears, and are handled by the same state rule of -Theorem~\ref{thm:tire-chromatic-polynomial-transfer}. -\end{remark} - \begin{theorem}[Tire treads form a rooted tree under face containment] \label{thm:tread-tree} Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ diff --git a/papers/three_color_restrictions_nested_tire_graphs/paper.aux b/papers/three_color_restrictions_nested_tire_graphs/paper.aux index 4a149ec..2f03a2b 100644 --- a/papers/three_color_restrictions_nested_tire_graphs/paper.aux +++ b/papers/three_color_restrictions_nested_tire_graphs/paper.aux @@ -13,40 +13,52 @@ \@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} +\citation{bauerfeld-nested-tire-decompositions} +\citation{tait-original} \@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. 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Every triangulation in this range admits at least one vertex source witnessing the conjecture.}}{7}{}\protected@file@percent } +\newlabel{tab:level-cycle-three-colour-counts}{{1}{7}} +\citation{bauerfeld-nested-tire-decompositions} +\citation{bauerfeld-nested-tire-decompositions} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The $5$-connected slice at $n \leq 24$}}{8}{}\protected@file@percent } +\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The $5$-connected triangulations at $14 \leq n \leq 24$ generated by \texttt {plantri -c5 -a}. 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PDF statistics: - 139 PDF objects out of 1000 (max. 8388607) - 84 compressed objects within 1 object stream + 150 PDF objects out of 1000 (max. 8388607) + 91 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 index bf5badc..cffaccc 100644 Binary files a/papers/three_color_restrictions_nested_tire_graphs/paper.pdf 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 index 31619ba..871dee4 100644 --- a/papers/three_color_restrictions_nested_tire_graphs/paper.tex +++ b/papers/three_color_restrictions_nested_tire_graphs/paper.tex @@ -114,7 +114,188 @@ 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. +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 proves that each tire tread has an outerplanar inner dual and that global colouring questions factor through local tread colourings together with compatibility along parent-child interfaces. + +\section{Single-tire colouring transfer} + +\begin{theorem}[Tait correspondence: 4-colorings of a tire vs 3-edge-colorings of its inner dual] +\label{thm:tait-tire} +Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph +(viewed as an annular triangulation of its tire tread $R$) and let +$\Gamma$ be its outerplanar inner dual, as supplied by +\cite{bauerfeld-nested-tire-decompositions}. Then +\[ + \#\bigl\{\text{proper $4$-vertex-colorings of } T\bigr\} \big/ |S_4| + \;=\; + \#\bigl\{\text{proper $3$-edge-colorings of } \Gamma\bigr\} \big/ |S_3|. +\] +That is, the number of $4$-vertex-colorings of $T$ up to permutation +of the colour set $\{0, 1, 2, 3\}$ equals the number of +$3$-edge-colorings of $\Gamma$ up to permutation of the colour set +$\{1, 2, 3\}$. +\end{theorem} + +\begin{proof} +The argument is the classical Tait correspondence +\cite{tait-original} adapted to the annular triangulation $T$. +Encode the four colours of a proper $4$-vertex-coloring $c \colon +V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$. For each interior +annular edge $e$ of $T$ (whose two incident faces both lie in +$F_{\mathrm{ann}}$, contributing a $\Gamma$-edge $e^*$), set +\[ + \chi^*(e^*) \;:=\; c(u) + c(v) \in \mathbb{Z}_2 \times \mathbb{Z}_2, + \qquad \text{where } u, v \text{ are the endpoints of } e. +\] +Since $c(u) \ne c(v)$, we have $\chi^*(e^*) \ne 00$, so $\chi^*$ +takes values in $\{01, 10, 11\}$, which we identify with the +$3$-edge-coloring palette $\{1, 2, 3\}$. + +\emph{Properness.} At each $\Gamma$-vertex $d_f$ corresponding to +an annular triangle $f = \{u, v, w\}$, the three incident +$\Gamma$-edges (one per cycle-edge of $f$) carry colours +$c(u) + c(v),\; c(v) + c(w),\; c(u) + c(w)$. These three elements +of $\mathbb{Z}_2 \times \mathbb{Z}_2$ sum to $0$ and are pairwise +distinct (their pairwise differences are +$c(u) - c(w),\; c(v) - c(u),\; c(w) - c(v)$, each nonzero), so +they form a permutation of $\{01, 10, 11\}$ --- a proper edge +colouring at $d_f$. + +\emph{Surjectivity onto cosets.} Given a proper $3$-edge-coloring +$\chi^*$ of $\Gamma$, the equation $c(u) + c(v) = \chi^*(e^*)$ +admits exactly $|\mathbb{Z}_2 \times \mathbb{Z}_2| = 4$ solutions +$c \colon V(T) \to \mathbb{Z}_2 \times \mathbb{Z}_2$ (a global +translation is the only freedom). Hence the map $c \mapsto +\chi^*$ is $4$-to-$1$. + +\emph{Count.} Therefore $\#\{\text{4-colorings of } T\} = 4 \cdot +\#\{\text{3-edge-colorings of } \Gamma\}$. Dividing by $|S_4| += 24$ on the left and $|S_3| = 6$ on the right (since $S_4$ acts +faithfully on the $4$-colorings and $S_3$ on the $3$-edge-colorings, +and the $4$-to-$1$ map respects the $S_4/S_3 \cong S_3$ quotient +via the natural surjection $S_4 \twoheadrightarrow S_3$) gives the +stated equality. +\end{proof} + +\begin{remark} +\label{rem:count-general-outerplanar} +Theorem~\ref{thm:tait-tire} reduces the $4$-colouring count of a +tire to the $3$-edge-coloring count of its outerplanar inner dual +$\Gamma$. For the cycle case $\Gamma \cong C_{\mu+\nu}$ (the spoke-only +case described in \cite{bauerfeld-nested-tire-decompositions}), the cycle +chromatic polynomial at $3$ colours gives +$2^{\mu+\nu} + 2 (-1)^{\mu+\nu}$. For an inner dual with one or more +non-crossing chords, the count depends on the chord structure, not just +on the pair (number of vertices, number of chords): two outerplanar +graphs with the same number of vertices and number of chords can have +different proper +$3$-edge-coloring counts depending on how the chords are arranged +(nested, sequential, sharing vertices, etc.). Every such count +can nevertheless be computed in linear time by tree-decomposition +methods, since outerplanar graphs have treewidth at most $2$ and +the edge-chromatic polynomial admits a deletion--contraction +recursion that respects the cycle-plus-chord structure. +\end{remark} + +\begin{definition}[Boundary-state chromatic transfer] +\label{def:boundary-state-transfer} +Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph. +Choose a cut along one annular edge if both boundaries are +non-degenerate; in the degenerate case make no cut. The tread becomes +a triangulated disk $\widetilde R$. Let +\[ + f_1, f_2, \ldots, f_m +\] +be any shelling order of the triangular faces of $\widetilde R$, i.e.\ +an order in which each initial union +$\widetilde R_i := f_1 \cup \cdots \cup f_i$ is a disk. Such an order +is obtained by taking an outerplanar embedding of the inner dual +$\Gamma$ from \cite{bauerfeld-nested-tire-decompositions} and repeatedly +removing an outer-face ear. + +For each $i$, let $A_i$ be the \emph{frontier}: the vertices of +$T$ incident to at least one processed face in $\widetilde R_i$ and to +at least one still-unprocessed constraint, where the unprocessed +constraints are the remaining annular faces together with any edge of +$O$ not yet tested by the transfer. A \emph{boundary state} on $A_i$ +is a partition $\pi$ of $A_i$ into colour classes, subject to the +condition that adjacent vertices of $T[A_i]$ lie in distinct blocks. +We write $r(\pi)$ for the number of blocks of $\pi$. +\end{definition} + +\begin{theorem}[Chromatic polynomial of a tire by frontier transfer] +\label{thm:tire-chromatic-polynomial-transfer} +For every tire graph $T$, the chromatic polynomial $P_T(q)$ is +computed by the following boundary-state dynamic program. + +Initialize the table at $i=0$ with the empty frontier state of weight +$1$. When the next triangular face +$f_i = \{x,y,z\}$ is attached, pass from states on $A_{i-1}$ to states +on $A_i$ as follows. +\begin{enumerate} +\item[(1)] Introduce any vertices of $f_i$ not already present in + $A_{i-1}$, assigning each such vertex either to an existing + colour block not containing one of its already-coloured + neighbours, or to a new block. +\item[(2)] Reject every assignment in which two adjacent vertices of + the triangle $f_i$ lie in the same block. Also reject every + assignment in which an edge of $O$ whose two endpoints have + now both appeared for the first time as a tested pair has + both endpoints in the same block. Thus chords and bridges of + the inner outerplanar graph are enforced exactly when their + second endpoint becomes visible to the transfer. +\item[(3)] Delete from the state every vertex no longer incident to an + unprocessed constraint. If deleting a vertex removes the last + representative of its colour block from the frontier, multiply + that transition by $1$; the colour has already been chosen. +\item[(4)] If a new vertex is assigned to a new colour block while the + current frontier state has $r$ colour blocks, multiply that + transition by $q-r$. If several new colour blocks are created + in the same triangle, the factors are + $(q-r)(q-r-1)\cdots$ in the order of creation. +\end{enumerate} +After $f_m$ is processed, the frontier is empty. The single remaining +weight is $P_T(q)$. +\end{theorem} + +\begin{proof} +The construction is the standard transfer for the chromatic polynomial, +specialized to the tire shelling. The frontier state records exactly +the equality pattern among colours that can still affect unprocessed +faces. Since colour names are irrelevant to the chromatic polynomial, +states are quotiented by the natural action of the symmetric group on +the colour set; a state with $r$ visible colour blocks can be extended +by a genuinely new colour in $q-r$ ways. + +Each transition accounts for all proper colourings of the enlarged +processed disk $\widetilde R_i$ that restrict to the resulting frontier +state, and accounts for none that violate an edge of the newly attached +triangle or untested edge of $O$. Vertices removed from the frontier +have no future incident unprocessed constraint, so their actual colour +names can no longer influence compatibility and may be forgotten. +Induction on $i$ therefore shows +that the table after step $i$ is precisely the orbit-count generating +function for proper colourings of $\widetilde R_i$ by frontier state. +At $i=m$ no vertices remain active, so the accumulated weight counts all +proper colourings of $T$. Because the weights are polynomials in $q$, +this count is the full chromatic polynomial. +\end{proof} + +\begin{remark}[Spoke-only transfer matrix] +\label{rem:spoke-only-chromatic-transfer} +In the spoke-only case with both boundaries simple cycles, the method +has a particularly small form. Cut the annulus along one spoke and +walk around the resulting strip. Each step adds one triangle sharing +an edge with the previous processed strip, so the frontier consists of +two or three consecutive boundary vertices. Up to colour permutation +there are only the possible equality patterns among those active +vertices, with adjacent vertices required to be distinct. The +chromatic polynomial is therefore the trace of a finite transfer matrix +whose entries are polynomials in $q$; the matrix depends only on the +local triangle type encountered while walking around the tread. Chords +or cut-vertices of $O$ enlarge the frontier only at the corresponding +outerplanar ears, and are handled by the same state rule of +Theorem~\ref{thm:tire-chromatic-polynomial-transfer}. +\end{remark} \begin{remark}[Motivation for level-cycle restrictions] \label{rem:level-cycle-motivation}