From afe4bf4859e5ba236ef6fc4952e0c7642f9d8c2b Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 23:14:58 -0400 Subject: [PATCH] coloring_nested_tire_graphs: define inner and outer spokes (Def 1.17) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds a new definition partitioning V(T'_{f'}) \ V(f') by geometric location relative to the face f': V_out(T'_{f'}) := { v in V(T'_{f'}) \ V(f') : v lies outside the closure of f' } = "outer spokes" V_in(T'_{f'}) := { v in V(T'_{f'}) \ V(f') : v lies inside the open region f' } = "inner spokes" These are well-defined because the boundary walk of f' is V(f') by definition, so no element of V(T'_{f'}) \ V(f') sits on ∂f'. In the spoke-only setting (T'_ann = C_{n+m}), the inner spokes of the inner face are the O-side non-annular dual vertices and the outer spokes are the source-side non-annular dual vertices (and vice-versa for the outer face). Paper stays at 10 pages. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 11 ++++---- papers/coloring_nested_tire_graphs/paper.log | 24 ++++++++--------- papers/coloring_nested_tire_graphs/paper.pdf | Bin 813184 -> 814471 bytes papers/coloring_nested_tire_graphs/paper.tex | 26 +++++++++++++++++++ 4 files changed, 43 insertions(+), 18 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index b9a54c6..27d2a3b 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -20,17 +20,18 @@ \newlabel{rem:tire-component-degenerate}{{1.11}{8}} \newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}} \newlabel{prop:edge-vertex-bijection}{{1.13}{8}} +\newlabel{rem:edge-vertex-corollary}{{1.14}{9}} +\newlabel{def:tire-annular-subgraph}{{1.15}{9}} +\newlabel{def:tire-annular-face-connector}{{1.16}{9}} +\newlabel{def:spokes}{{1.17}{9}} +\newlabel{rem:facial-dual-spoke-only}{{1.18}{9}} \bibcite{bauerfeld-pds}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:edge-vertex-corollary}{{1.14}{9}} -\newlabel{def:tire-annular-subgraph}{{1.15}{9}} -\newlabel{def:tire-annular-face-connector}{{1.16}{9}} -\newlabel{rem:facial-dual-spoke-only}{{1.17}{9}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{10}{}\protected@file@percent } \newlabel{fig:facial-dual-choices}{{5}{10}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent } \gdef \@abspage@last{10} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index baa2443..2fd8af9 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 23:06 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 23:14 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -228,28 +228,26 @@ LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: Reference `def:dual' on page 9 undefined on input line 575. - +[9] + File: notes/fig_facial_dual_choices.png Graphic file (type png) Package pdftex.def Info: notes/fig_facial_dual_choices.png used on input line -628. +654. (pdftex.def) Requested size: 360.0pt x 143.50418pt. - -LaTeX Warning: `h' float specifier changed to `ht'. - -[9] [10 <./notes/fig_facial_dual_choices.png>] (./paper.aux) + [10 <./notes/fig_facial_dual_choices.png>] (./paper.aux) LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: - 3041 strings out of 478268 - 43057 string characters out of 5846347 - 344281 words of memory out of 5000000 - 21084 multiletter control sequences out of 15000+600000 + 3042 strings out of 478268 + 43069 string characters out of 5846347 + 344292 words of memory out of 5000000 + 21085 multiletter control sequences out of 15000+600000 475666 words of font info for 53 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,8n,76p,1079b,316s stack positions out of 10000i,1000n,20000p,200000b,200000s + 69i,14n,76p,1079b,316s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -266,7 +264,7 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb> -Output written on paper.pdf (10 pages, 813184 bytes). +Output written on paper.pdf (10 pages, 814471 bytes). 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