From 688b944f956cd4ba05bcf28598802afc60473d30 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 15:31:59 -0400 Subject: [PATCH] coloring_nested_tire_graphs: firm up Lemma 1.7 with explicit manifold/boundary hypotheses MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds two explicit hypotheses (R1) and (R2) to Lemma 1.7 (tire-component lemma) and tightens the proof to use them precisely: (R1) R_{C'} is a topological 2-manifold with boundary; equivalently, at every v ∈ V_{C'} the depth-d faces of F_{C'} incident to v form a single contiguous arc in v's rotation in Π_G. (R2) R_{C'} has at most two boundary components. These rule out, respectively, pinch points (where C' wraps around a vertex via a global path of depth-d faces, producing a non-manifold region) and multi-hole topology (a "pair of pants" or worse, which can occur when several disjoint depth->d lobes sit inside one depth-d component). The proof is reorganised into labelled steps: 1. Outerplanarity of the two level parts (via Lemma 2.6 of [1]). 2. Layer containment V_{C'} ⊆ L_d ∪ L_{d+1}. 3. Boundary edges are monochromatic in level (full case analysis on the third vertex of the outside face f', using the bounded-step property of δ). 4. Boundary components are simple cycles (uses R1: locally at any boundary point R_{C'} is a half-disk, so the boundary walk visits each vertex once). 5. Topological type: planarity + R1 + R2 forces disk or annulus via the classification of compact orientable 2-manifolds with boundary. 6. Tire structure: identifies the two boundary parts as the level-d and level-(d+1) induced subgraphs, in either order. Adds Remark 1.10 documenting when (R1) and (R2) hold or fail: - (R1) fails iff there is a pinch vertex whose cyclic level sequence around it enters and leaves {d, d+1} more than once. - (R2) fails iff R_{C'} encloses two or more disjoint depth->d sub-regions (the depth- --- papers/coloring_nested_tire_graphs/paper.aux | 7 +- papers/coloring_nested_tire_graphs/paper.log | 50 +++--- papers/coloring_nested_tire_graphs/paper.pdf | Bin 464573 -> 477312 bytes papers/coloring_nested_tire_graphs/paper.tex | 177 ++++++++++++++----- 4 files changed, 161 insertions(+), 73 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 022f824..1a99d60 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -9,12 +9,13 @@ \@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}{4}} \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} +\newlabel{rem:R1-R2-when}{{1.9}{5}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent } +\gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 6625f02..e08f9da 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 15:23 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 15:31 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -210,35 +210,35 @@ Package pdftex.def Info: fig_tire_example.png used on input line 154. 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