From a61bdcfd69bcf1ec4b8c56ca40cbe653318ca0dc Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 15:17:28 -0400 Subject: [PATCH] coloring_nested_tire_graphs: prove Lemma 1.7 using Lemma 2.6 of plane_depth_sequencing MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit - Adds a proof sketch to Lemma 1.7 (tire-component lemma). The outerplanarity step cites Lemma 2.6 of `bauerfeld-pds` (the Plane Depth Sequencing manuscript), which proves that for any level source the subgraph induced on a single depth level is outerplanar. The proof notes that the cited result, originally stated for an outer-cycle source, extends verbatim to an arbitrary level source by treating S as the depth-0 set. - The remainder of the proof: layer containment forces V_{C'} ⊆ L_d ∪ L_{d+1}; a face-by-face boundary analysis shows ∂R_{C'} is monochromatic in level; connectivity of C' rules out higher genus; the resulting one or two closed boundary walks give the tire graph's two boundary parts (with the degenerate case at the BFS endpoints). - Adds a thebibliography block with one entry for the cited paper. The paper grows from 3 to 4 pages. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 17 ++++--- papers/coloring_nested_tire_graphs/paper.log | 51 ++++++++++--------- papers/coloring_nested_tire_graphs/paper.pdf | Bin 449333 -> 464467 bytes papers/coloring_nested_tire_graphs/paper.tex | 50 ++++++++++++++++++ 4 files changed, 86 insertions(+), 32 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 44ee562..022f824 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -5,13 +5,16 @@ \newlabel{fig:dual-depth}{{1}{2}} \newlabel{def:tire-graph}{{1.5}{2}} \newlabel{rem:tire-counts}{{1.6}{2}} -\newlabel{tocindent-1}{0pt} -\newlabel{tocindent0}{0pt} -\newlabel{tocindent1}{17.77782pt} -\newlabel{tocindent2}{0pt} -\newlabel{tocindent3}{0pt} +\citation{bauerfeld-pds} \@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{lem:tire-component}{{1.7}{3}} -\newlabel{rem:tire-component-degenerate}{{1.8}{3}} -\gdef \@abspage@last{3} +\bibcite{bauerfeld-pds}{1} +\newlabel{tocindent-1}{0pt} +\newlabel{tocindent0}{12.7778pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{0pt} +\newlabel{tocindent3}{0pt} +\newlabel{rem:tire-component-degenerate}{{1.8}{4}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent } +\gdef \@abspage@last{4} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 4c7d95c..b6c5179 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) 25 MAY 2026 15:00 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 15:17 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -210,34 +210,35 @@ Package pdftex.def Info: fig_tire_example.png used on input line 154. 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