From 787d9f0caf240fdc2d66e88f89f219ee484e9e76 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 03:04:31 -0400 Subject: [PATCH] coloring_nested_tire_graphs: drop count-formula theorem, keep remark MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Removed Theorem 1.16 (the count formula for spoke-only and single-chord cases). Folded the cycle formula 2^n + 2(-1)^n into the surviving remark so the only retained content is the structural observation: - Tait reduces 4-coloring count to 3-edge-coloring count of Γ. - For Γ ≅ C_n (spoke-only): cycle chromatic polynomial gives 2^n + 2(-1)^n. - For Γ with chords, the count depends on chord structure (nested vs. sequential etc.), not just (n, k). - Always computable in linear time via tree decomposition (outerplanar has treewidth ≤ 2). Page count: 12 → 11. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 10 +-- papers/coloring_nested_tire_graphs/paper.log | 12 +-- papers/coloring_nested_tire_graphs/paper.pdf | Bin 623067 -> 618557 bytes papers/coloring_nested_tire_graphs/paper.tex | 73 ++++--------------- 4 files changed, 25 insertions(+), 70 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index b6be073..48210bc 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -26,11 +26,10 @@ \newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}} \newlabel{rem:bridge-case-theta}{{1.14}{9}} \citation{tait-original} -\citation{bauerfeld-nested-tire-duals} \newlabel{thm:tait-tire}{{1.15}{10}} -\newlabel{thm:count-formula}{{1.16}{10}} -\newlabel{rem:count-general-outerplanar}{{1.17}{11}} -\newlabel{thm:tread-tree}{{1.18}{11}} +\newlabel{rem:count-general-outerplanar}{{1.16}{10}} +\newlabel{thm:tread-tree}{{1.17}{10}} +\newlabel{rem:tree-multiple-children}{{1.18}{11}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} @@ -39,7 +38,6 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:tree-multiple-children}{{1.19}{12}} -\newlabel{rem:tree-coloring-factorisation}{{1.20}{12}} +\newlabel{rem:tree-coloring-factorisation}{{1.19}{12}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent } \gdef \@abspage@last{12} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index a99f1ec..d2f4a7e 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) 27 MAY 2026 02:57 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 03:04 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -513,10 +513,10 @@ LaTeX Warning: `h' float specifier changed to `ht'. 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