From 4a2df75773d62192541e609c4e26b8989a5bdf00 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 22:37:03 -0400 Subject: [PATCH] coloring_nested_tire_graphs: add Definition 1.15 (partial tire facial dual) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds a new definition formalizing the "partial tire facial dual" T'_{f'}: (i) Annular dual subgraph T'_ann := G'[{d_f : f ∈ F_ann}], with planar embedding inherited from G' (where G' is the inner planar dual of the maximal planar G). (ii) For each face f' of T'_ann in its inherited embedding, T'_{f'} := closed G'-neighborhood of V(f') together with every G'-edge incident to V(f'). Adds a remark noting that in the spoke-only case T'_ann = Γ ≅ C_{n+m} has two faces (both with V(f') = all interior dual vertices), and T'_{f'} recovers the planar dual of T when G is the tire plus one source-side and one O-side face. Paper stays at 9 pages. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 2 + papers/coloring_nested_tire_graphs/paper.log | 55 +++++++++++-------- papers/coloring_nested_tire_graphs/paper.pdf | Bin 746924 -> 752040 bytes papers/coloring_nested_tire_graphs/paper.tex | 53 ++++++++++++++++++ 4 files changed, 86 insertions(+), 24 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 14180d5..9f155fe 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -27,5 +27,7 @@ \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} \newlabel{rem:edge-vertex-corollary}{{1.14}{9}} +\newlabel{def:partial-tire-facial-dual}{{1.15}{9}} +\newlabel{rem:facial-dual-spoke-only}{{1.16}{9}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent } \gdef \@abspage@last{9} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 187d619..ba18c94 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 20:52 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 22:36 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -224,33 +224,40 @@ Package pdftex.def Info: fig_partial_tire_dual_bridge.png used on input line 2 LaTeX Warning: `h' float specifier changed to `ht'. 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