From 9b4100e143b282d3b3bb09cffc3ee94c8598f117 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 01:01:32 -0400 Subject: [PATCH] papers: name R the "tire tread" in Definition 1.5 Foundational paper: Definition 1.5 (Tire graph) now explicitly names the closed planar region R bounded by B_out and B_in the "tire tread of T". Remark 1.6 and the abstract updated to use the new term. Dual paper: places that referred to R as "the closed annular region" or "the annular region" updated to use "tire tread" for consistency: - Definition 1.1 (Partial tire dual) - Caption of figure on partial tire dual example - Two places inside the proof of Proposition 1.2 "annular edges" (E_ann) and "annular faces" (F_ann) kept as-is since they're established notation; the tread is the region they triangulate. Co-Authored-By: Claude Opus 4.7 --- .../paper.aux | 2 +- .../paper.log | 4 +- .../paper.pdf | Bin 605784 -> 605799 bytes .../paper.tex | 10 ++--- papers/coloring_nested_tire_graphs/paper.log | 8 ++-- papers/coloring_nested_tire_graphs/paper.pdf | Bin 456808 -> 457250 bytes papers/coloring_nested_tire_graphs/paper.tex | 35 +++++++++--------- 7 files changed, 30 insertions(+), 29 deletions(-) diff --git a/papers/coloring_nested_tire_dual_graphs/paper.aux b/papers/coloring_nested_tire_dual_graphs/paper.aux index 77575ce..edb0823 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.aux +++ b/papers/coloring_nested_tire_dual_graphs/paper.aux @@ -8,7 +8,7 @@ \citation{bauerfeld-nested-tires} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } \newlabel{def:partial-tire-dual}{{1.1}{1}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{2}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the tire tread (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{2}{}\protected@file@percent } \newlabel{fig:partial-tire-dual-example}{{1}{2}} \newlabel{prop:partial-tire-dual-structure}{{1.2}{2}} \citation{bauerfeld-nested-tires} diff --git a/papers/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index 4d55b1f..f0740d3 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.log +++ b/papers/coloring_nested_tire_dual_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 00:58 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:01 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -258,7 +258,7 @@ live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb> -Output written on paper.pdf (9 pages, 605784 bytes). +Output written on paper.pdf (9 pages, 605799 bytes). 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a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -50,9 +50,9 @@ triangulations. A \emph{level source} of a triangulation $G$ induces a BFS layering of $G$, which in turn endows the inner planar dual $G'$ with a \emph{dual depth} grading. We isolate the basic object of study --- the \emph{tire graph} $T$, a plane graph -whose outer and inner boundaries bound an annular region -triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$ --- and -record its face/edge counts. +whose outer and inner boundaries bound the \emph{tire tread} $R$, +a closed region triangulated by the \emph{annular edges} +$E_{\mathrm{ann}}$ --- and record its face/edge counts. \end{abstract} \maketitle @@ -150,20 +150,21 @@ The vertex and edge sets of $T$ are E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}}, \] where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the -property that, in the plane embedding of $T$, the closed planar region -$R$ bounded externally by $B_{\mathrm{out}}$ and internally by -$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose -union is $R$. +property that, in the plane embedding of $T$, the closed planar +region $R$ bounded externally by $B_{\mathrm{out}}$ and internally +by $B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ +whose union is $R$. We call $R$ the \emph{tire tread} of $T$. When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected, -$R$ is a closed annulus. More generally, $R$ is a closed planar -region that may fail to be a $2$-manifold at cut-vertices of $O$ (where -two ``lobes'' of the depth-$d$ region meet at a single vertex); the -inner boundary $B_{\mathrm{in}}$ is then a non-simple closed walk that -visits the cut-vertex multiple times. The relaxed definition -accommodates outerplanar inner graphs with bridges, cut-vertices, or -multiple connected components. When either boundary is degenerate, -$R$ is a closed disk with that vertex as apex. +the tread is a closed annulus. More generally, $R$ is a closed +planar region that may fail to be a $2$-manifold at cut-vertices of +$O$ (where two ``lobes'' of the depth-$d$ region meet at a single +vertex); the inner boundary $B_{\mathrm{in}}$ is then a non-simple +closed walk that visits the cut-vertex multiple times. The relaxed +definition accommodates outerplanar inner graphs with bridges, +cut-vertices, or multiple connected components. When either +boundary is degenerate, the tread is a closed disk with that vertex +as apex. \end{definition} \begin{figure}[h] @@ -182,8 +183,8 @@ triangular faces.} \begin{remark} \label{rem:tire-counts} Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By -Euler's formula on the annular (resp.\ disk) region $R$, the tire graph -has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$ +Euler's formula on the tire tread $R$, the tire graph has $m + k$ +triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$ annular edges when neither boundary is degenerate; when exactly one boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$ triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.