From e4216ec7a21293501f86796ae22796fff09cdc91 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 18:00:05 -0400 Subject: [PATCH] coloring_nested_tire_graphs: add Definition 1.7 (Partial tire dual) + structure proposition MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds Definition 1.7 (Partial tire dual) formalising the user's construction: for a tire graph T with annular face set F_{ann}, the partial tire dual D(T) has - Interior vertices d_f for each annular face f, - Leaf vertices for each edge of B_out and each occurrence of an edge on the boundary walk B_in (so cut-vertices/bridges of O contribute multiple leaves), - Interior dual edges for each annular edge incident to two annular faces, - Leaf edges from d_f to the corresponding leaf for each boundary edge of the annular region. Adds Proposition 1.8 showing that when the annular triangulation is spoke-only (i.e. every annular edge has one endpoint on B_out and one on B_in) and O is 2-connected, each annular face has exactly 1 boundary edge + 2 interior annular edges. Consequently each interior vertex d_f has degree 3 = 2 (cycle) + 1 (leaf), and the induced subgraph on {d_f} is a single cycle of length n + m. D(T) is then isomorphic to the corona C_{n+m} ∘ K_1 -- a cycle of length n+m with one leaf attached to each cycle vertex; |V(D(T))| = |E(D(T))| = 2(n+m). Subsequent numbering shifted: Proposition (Source-side simple-cycle property) is now 1.9; Lemma (Tire-component) is now 1.10; Remarks shift to 1.11 and 1.12. All cross-references are by label, so they update automatically. Paper grows from 6 to 7 pages. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 14 ++-- papers/coloring_nested_tire_graphs/paper.log | 57 +++++++------- papers/coloring_nested_tire_graphs/paper.pdf | Bin 492830 -> 499770 bytes papers/coloring_nested_tire_graphs/paper.tex | 76 +++++++++++++++++++ 4 files changed, 111 insertions(+), 36 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 7f0b0f7..25d708b 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -7,17 +7,19 @@ \@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{prop:no-level-d-pinch}{{1.7}{3}} +\newlabel{def:partial-tire-dual}{{1.7}{3}} +\newlabel{prop:partial-tire-dual-structure}{{1.8}{4}} +\newlabel{prop:no-level-d-pinch}{{1.9}{4}} \citation{bauerfeld-pds} -\newlabel{lem:tire-component}{{1.8}{4}} +\newlabel{lem:tire-component}{{1.10}{5}} \citation{bauerfeld-pds} -\newlabel{rem:tire-component-degenerate}{{1.9}{5}} +\newlabel{rem:tire-component-degenerate}{{1.11}{6}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{6}} \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-no-extra-hypotheses}{{1.10}{6}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent } -\gdef \@abspage@last{6} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent } +\gdef \@abspage@last{7} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 699a6c6..93cecae 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 17:03 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 17:59 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -207,39 +207,36 @@ File: fig_tire_example.png Graphic file (type png) Package pdftex.def Info: fig_tire_example.png used on input line 160. 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