From 6413560a7b5c361de1d3e2d81a9eb754c5ab3ac2 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 03:32:44 -0400 Subject: [PATCH] coloring_nested_tire_graphs: conjecture sketch on universal nesting MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit NEW Conjecture 1.19 (universal nesting of tire-tread trees, sketch): For any two rooted trees of tire treads T_1 = T(G_1, S_1) and T_2 = T(G_2, S_2), T_1 NESTS into T_2: Choose any tire T in T_2 and any non-trivial bounded face f of its inner outerplanar graph O^(T). Then there exists a maximal planar graph G̃ with level source S̃ such that: (N1) T(G̃, S̃) contains T_2 as a sub-tree. (N2) The sub-tree rooted at the new child of T at face f is isomorphic to T_1. Informally: any tree of tire treads can be inserted into any non-trivial face slot of any other tree of tire treads. The class of trees of tire treads is closed under composition by face-slot insertion. Followed by Remark 1.20 motivating the conjecture: - Compositional colourability: if 4-colourability of G̃ follows from 4-colourability of G_1, G_2 via parent-child consistency (Remark 1.18 / former tree-coloring-factorisation), then 4CT propagates through nesting. A min 4CT counterexample would have to be irreducible under such nesting. - Universality: trees of tire treads become a "term algebra" for decomposing plane triangulations; coloring arguments can be inductive on this algebra. Open subquestions in remark: - Precise notion of "isomorphic as rooted trees of tire treads" (combinatorial vs geometric vs up to embedding). - Constructive description of G̃ from G_1, G_2, f. - Compatibility with Birkhoff's internally 6-connected condition. Page count: 12 → ~13. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 8 ++- papers/coloring_nested_tire_graphs/paper.log | 70 +++++++++---------- papers/coloring_nested_tire_graphs/paper.pdf | Bin 618557 -> 623232 bytes papers/coloring_nested_tire_graphs/paper.tex | 69 ++++++++++++++++++ 4 files changed, 109 insertions(+), 38 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 48210bc..ab4e642 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -30,6 +30,9 @@ \newlabel{rem:count-general-outerplanar}{{1.16}{10}} \newlabel{thm:tread-tree}{{1.17}{10}} \newlabel{rem:tree-multiple-children}{{1.18}{11}} +\newlabel{rem:tree-coloring-factorisation}{{1.19}{12}} +\newlabel{conj:universal-nesting}{{1.20}{12}} +\newlabel{rem:nesting-motivation}{{1.21}{12}} \bibcite{tait-original}{1} \bibcite{bauerfeld-depth}{2} \bibcite{bauerfeld-nested-tire-duals}{3} @@ -38,6 +41,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:tree-coloring-factorisation}{{1.19}{12}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent } -\gdef \@abspage@last{12} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{13}{}\protected@file@percent } +\gdef \@abspage@last{13} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index d2f4a7e..64faef6 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 03:04 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 03:32 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -511,45 +511,45 @@ Package pdftex.def Info: fig_tire_example.png used on input line 179. 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