From f868de4f671f93bfe4cae089067005d5525d1f11 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 02:40:20 -0400 Subject: [PATCH] coloring_nested_tire_graphs: theorem on tire-tread rooted tree structure MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit NEW Theorem 1.15: The tire treads from a single-vertex level source S = {v_0} form a rooted tree T(G, S) under face containment. Statement: - Root: the depth-0 tire tread T_0 with degenerate outer boundary {v_0} (the apex tire, B_out = {v_0}). - Parent: for any tire tread T_c at depth d ≥ 1, the unique parent T_p at depth d-1 is the tire whose inner outerplanar graph O^(p) has B_out^(c) as one of its bounded faces. Equivalently, R_c lies inside this bounded face of O^(p). - Children: bijection with bounded faces of O^(p) whose interior contains depth-≥(d+2) vertices. Proof structure: 1. Root well-defined: G'_0 is connected (fan around v_0), so unique component → unique T_0. 2. Existence of parent: faces immediately outside B_out^(c) on the S-side have depth d-1, lie in some component of G'_{d-1}. 3. Uniqueness: by Proposition 1.7 (source-side simple-cycle property), B_out^(c) is a simple cycle, and the depth-(d-1) faces around it form a single contiguous arc in the dual, hence one component → unique parent. 4. Children description: bounded faces of O^(p) are in bijection with deeper component-tires. 5. Tree property: parent map strictly decreases depth, hence no cycles, hence rooted tree. Plus two clarifying remarks: - Remark 1.16: multiple children iff O^(p) has multiple bounded faces with non-trivial interiors. Spoke-only case → exactly one child. - Remark 1.17: combined with Theorem 1.9 (partition) and Theorem 1.12 (outerplanar inner dual), any coloring problem on G factors through: • local outerplanar coloring on each tread, • parent-child consistency along shared B_out^(c) cycles. This is the structural setup for the chain-pigeonhole program. Page count: 10 → 11. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 7 +- papers/coloring_nested_tire_graphs/paper.log | 70 +++++------ papers/coloring_nested_tire_graphs/paper.pdf | Bin 594243 -> 603044 bytes papers/coloring_nested_tire_graphs/paper.tex | 116 +++++++++++++++++++ 4 files changed, 158 insertions(+), 35 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 4029993..7329362 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -25,6 +25,7 @@ \newlabel{fig:inner-dual-annulus-case}{{4}{9}} \newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}} \newlabel{rem:bridge-case-theta}{{1.14}{9}} +\newlabel{thm:tread-tree}{{1.15}{10}} \bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-nested-tire-duals}{2} \newlabel{tocindent-1}{0pt} @@ -32,5 +33,7 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent } -\gdef \@abspage@last{10} +\newlabel{rem:tree-multiple-children}{{1.16}{11}} +\newlabel{rem:tree-coloring-factorisation}{{1.17}{11}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{11}{}\protected@file@percent } +\gdef \@abspage@last{11} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index a7d0e98..f23232a 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 02:24 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 02:40 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -511,43 +511,47 @@ Package pdftex.def Info: fig_tire_example.png used on input line 179. 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