From 65f79f2e65423d1fae307cda0e55d8f133022518 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 00:54:53 -0400 Subject: [PATCH] papers: split coloring_nested_tire foundations into separate paper MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit NEW PAPER: papers/coloring_nested_tire_graphs/ ("Coloring Nested Tire Graphs", 5 pages). Contains foundational definitions 1.1 through 1.7 from the dual paper, plus the four illustrative figures: - 1.1 Level source - 1.2 Levels - 1.3 Dual (with label def:dual added — was missing in original) - 1.4 Dual depth - 1.5 Tire graph - 1.6 Remark (tire counts) - 1.7 Partial tire dual Also: the dual-depth figure, the tire-example figure, and both partial-tire-dual figures (vanilla + bridge case). MODIFIED: papers/coloring_nested_tire_dual_graphs/paper.tex now a follow-up: - Abstract recasts the paper as building on the foundational paper. - Intro no longer recapitulates definitions; lists them as citations to the new paper. - Removes definitions 1.1-1.7 and their figures (now in foundational paper). - Internal \ref{...} to removed labels converted to \cite[Definition N.M]{bauerfeld-nested-tires}. - Bibliography adds the new paper as a reference. - Renumbering: theorems/propositions now start at 1.1 (formerly 1.8). Paper down from 14 to 8 pages. Both papers compile cleanly with no broken references. Co-Authored-By: Claude Opus 4.7 --- .../paper.aux | 64 ++-- .../paper.log | 114 +++---- .../paper.pdf | Bin 837347 -> 362539 bytes .../paper.tex | 246 +++------------ .../fig_dual_depth.png | Bin 0 -> 255786 bytes .../fig_partial_tire_dual.png | Bin 0 -> 150413 bytes .../fig_partial_tire_dual_bridge.png | Bin 0 -> 146308 bytes .../fig_tire_example.png | Bin 0 -> 104494 bytes papers/coloring_nested_tire_graphs/paper.aux | 26 ++ papers/coloring_nested_tire_graphs/paper.log | 255 ++++++++++++++++ papers/coloring_nested_tire_graphs/paper.pdf | Bin 0 -> 711472 bytes papers/coloring_nested_tire_graphs/paper.tex | 286 ++++++++++++++++++ 12 files changed, 687 insertions(+), 304 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/fig_dual_depth.png create mode 100644 papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png create mode 100644 papers/coloring_nested_tire_graphs/fig_partial_tire_dual_bridge.png create mode 100644 papers/coloring_nested_tire_graphs/fig_tire_example.png create mode 100644 papers/coloring_nested_tire_graphs/paper.aux create mode 100644 papers/coloring_nested_tire_graphs/paper.log create mode 100644 papers/coloring_nested_tire_graphs/paper.pdf create mode 100644 papers/coloring_nested_tire_graphs/paper.tex diff --git a/papers/coloring_nested_tire_dual_graphs/paper.aux b/papers/coloring_nested_tire_dual_graphs/paper.aux index a258f40..4696eed 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.aux +++ b/papers/coloring_nested_tire_dual_graphs/paper.aux @@ -1,41 +1,43 @@ \relax +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } -\newlabel{def:dual-depth}{{1.4}{1}} -\@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.}}{2}{}\protected@file@percent } -\newlabel{fig:dual-depth}{{1}{2}} -\newlabel{def:tire-graph}{{1.5}{2}} -\@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.}}{3}{}\protected@file@percent } -\newlabel{fig:tire-example}{{2}{3}} -\newlabel{rem:tire-counts}{{1.6}{3}} -\newlabel{def:partial-tire-dual}{{1.7}{3}} -\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.8\hbox {}.}}{4}{}\protected@file@percent } -\newlabel{fig:partial-tire-dual-example}{{3}{4}} -\newlabel{prop:partial-tire-dual-structure}{{1.8}{4}} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.8\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{5}{}\protected@file@percent } -\newlabel{fig:partial-tire-dual-bridge}{{4}{5}} -\newlabel{prop:no-level-d-pinch}{{1.9}{5}} +\newlabel{prop:partial-tire-dual-structure}{{1.1}{1}} +\citation{bauerfeld-nested-tires} +\newlabel{prop:no-level-d-pinch}{{1.2}{2}} +\citation{bauerfeld-nested-tires} \citation{bauerfeld-depth} -\newlabel{lem:tire-component}{{1.10}{6}} +\citation{bauerfeld-nested-tires} +\newlabel{lem:tire-component}{{1.3}{3}} \citation{bauerfeld-depth} -\newlabel{rem:tire-component-degenerate}{{1.11}{8}} -\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}} -\newlabel{prop:edge-vertex-bijection}{{1.13}{8}} -\newlabel{rem:edge-vertex-corollary}{{1.14}{9}} -\newlabel{def:tire-annular-subgraph}{{1.15}{9}} -\newlabel{def:tire-annular-face-connector}{{1.16}{9}} -\newlabel{def:spokes}{{1.17}{9}} -\newlabel{rem:facial-dual-spoke-only}{{1.18}{9}} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{10}{}\protected@file@percent } -\newlabel{fig:facial-dual-choices}{{5}{10}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{10}{}\protected@file@percent } -\newlabel{sec:latin-conjecture}{{2}{10}} +\citation{bauerfeld-nested-tires} +\newlabel{rem:tire-component-degenerate}{{1.4}{4}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.5}{4}} +\newlabel{prop:edge-vertex-bijection}{{1.6}{4}} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\newlabel{rem:edge-vertex-corollary}{{1.7}{5}} +\newlabel{def:tire-annular-subgraph}{{1.8}{5}} +\newlabel{def:tire-annular-face-connector}{{1.9}{5}} +\newlabel{def:spokes}{{1.10}{6}} +\newlabel{rem:facial-dual-spoke-only}{{1.11}{6}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{6}{}\protected@file@percent } +\newlabel{sec:latin-conjecture}{{2}{6}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{7}{}\protected@file@percent } +\newlabel{fig:facial-dual-choices}{{1}{7}} +\newlabel{conj:latin}{{2.1}{7}} +\newlabel{conj:chain-latin}{{2.2}{7}} \bibcite{bauerfeld-depth}{1} +\bibcite{bauerfeld-nested-tires}{2} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{conj:latin}{{2.1}{11}} -\newlabel{conj:chain-latin}{{2.2}{11}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{11}{}\protected@file@percent } -\gdef \@abspage@last{11} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent } +\gdef \@abspage@last{8} diff --git a/papers/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index 82fbb25..b9e1bdf 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 00:44 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:54 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -191,52 +191,20 @@ Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) - -File: fig_dual_depth.png Graphic file (type png) - -Package pdftex.def Info: fig_dual_depth.png used on input line 107. -(pdftex.def) Requested size: 251.9989pt x 237.67276pt. - - -LaTeX Warning: `h' float specifier changed to `ht'. - [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -[2 <./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 161. -(pdftex.def) Requested size: 280.79956pt x 188.56097pt. - [3 <./fig_tire_example.png>] - -File: fig_partial_tire_dual.png Graphic file (type png) - -Package pdftex.def Info: fig_partial_tire_dual.png used on input line 226. -(pdftex.def) Requested size: 280.79956pt x 233.36552pt. - -File: fig_partial_tire_dual_bridge.png Graphic file (type png) - -Package pdftex.def Info: fig_partial_tire_dual_bridge.png used on input line 2 -41. -(pdftex.def) Requested size: 306.0022pt x 204.59406pt. - - -LaTeX Warning: `h' float specifier changed to `ht'. - -[4 <./fig_partial_tire_dual.png>] [5 <./fig_partial_tire_dual_bridge.png>] -[6] [7] [8] - -LaTeX Warning: Reference `def:dual' on page 9 undefined on input line 577. - -[9] - +[2] [3] [4] [5] + File: notes/fig_facial_dual_choices.png Graphic file (type png) Package pdftex.def Info: notes/fig_facial_dual_choices.png used on input line -656. +495. 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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.}}{2}{}\protected@file@percent } +\newlabel{fig:dual-depth}{{1}{2}} +\newlabel{def:tire-graph}{{1.5}{2}} +\@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.}}{3}{}\protected@file@percent } +\newlabel{fig:tire-example}{{2}{3}} +\newlabel{rem:tire-counts}{{1.6}{3}} +\newlabel{def:partial-tire-dual}{{1.7}{3}} +\citation{bauerfeld-nested-tire-duals} +\bibcite{bauerfeld-depth}{1} +\bibcite{bauerfeld-nested-tire-duals}{2} +\newlabel{tocindent-1}{0pt} +\newlabel{tocindent0}{12.7778pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{0pt} +\newlabel{tocindent3}{0pt} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ analysed in the companion paper\nonbreakingspace \cite {bauerfeld-nested-tire-duals}.}}{4}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-example}{{3}{4}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of the spoke-only case, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{5}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-bridge}{{4}{5}} +\gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log new file mode 100644 index 0000000..d625a80 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -0,0 +1,255 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:54 +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 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