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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PDF statistics: - 108 PDF objects out of 1000 (max. 8388607) - 63 compressed objects within 1 object stream + 113 PDF objects out of 1000 (max. 8388607) + 67 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 11 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index f20f5c6..4aecf0d 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 4ad81c7..f01ecb9 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -178,6 +178,82 @@ boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$ triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$. \end{remark} +\begin{definition}[Partial tire dual] +\label{def:partial-tire-dual} +Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in +the sense of Definition~\ref{def:tire-graph}, and let $F_{\mathrm{ann}}$ +denote the set of triangular faces of $T$ in the closed annular region +between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$. The \emph{partial +tire dual} of $T$, written $D(T)$, is the graph defined as follows. + +\emph{Vertices.} +\begin{enumerate} + \item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an + \emph{interior vertex} $d_f$ of $D(T)$. + \item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a + \emph{leaf vertex} $\ell_e^{\mathrm{out}}$. + \item[(V3)] For each occurrence of an edge in the closed walk + $B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$), + a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$. (When $O$ is + $2$-connected each edge appears once; cut-vertices and + bridges of $O$ may cause an edge or vertex to appear more + than once.) +\end{enumerate} + +\emph{Edges.} +\begin{enumerate} + \item[(E1)] For each edge $e \in E(T)$ whose two incident faces both + lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}), + one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where + $f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces + incident to $e$. + \item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge + $\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where + $f \in F_{\mathrm{ann}}$ is the unique annular face incident + to $e$. The leaf $\ell_e^{\mathrm{out}}$ has degree $1$. + \item[(E3)] For each occurrence of $e$ on the boundary walk + $B_{\mathrm{in}}$, one edge + $\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where + $f \in F_{\mathrm{ann}}$ is the annular face incident to $e$ + on the side of that occurrence. The leaf + $\ell_e^{\mathrm{in}}$ has degree $1$. +\end{enumerate} +\end{definition} + +\begin{proposition}[Structure of $D(T)$ when the annular triangulation +is spoke-only] +\label{prop:partial-tire-dual-structure} +Suppose $B_{\mathrm{out}}$ is a simple cycle of length $n$, $O$ is a +$2$-connected outerplanar graph whose outer-face cycle +$B_{\mathrm{in}}$ has length $m$, and $E_{\mathrm{ann}}$ consists only +of \emph{spokes} (edges with one endpoint in $V(B_{\mathrm{out}})$ and +one in $V(B_{\mathrm{in}})$). Then each face $f \in F_{\mathrm{ann}}$ +has exactly one boundary edge (on $B_{\mathrm{out}}$ or +$B_{\mathrm{in}}$) and two interior annular edges, and consequently +$D(T)$ is isomorphic to the corona graph $C_{n+m} \circ K_1$: a cycle +of length $n + m$ on the interior vertices $\{d_f\}$, with one leaf +attached to each cycle vertex. + +In particular, $|V(D(T))| = 2(n+m)$ and $|E(D(T))| = 2(n+m)$. +\end{proposition} + +\begin{proof} +Each annular triangle $f$ in a spoke-only triangulation has the form +$\{x, y, z\}$ with $x \in V(B_{\mathrm{out}})$, $y \in V(B_{\mathrm{in}})$, +and $z$ also in $V(B_{\mathrm{out}}) \cup V(B_{\mathrm{in}})$. Of its +three edges, the one between the two same-side vertices +($x$-$z$ if both on $B_{\mathrm{out}}$, or $y$-$z$ if both on +$B_{\mathrm{in}}$) is a boundary edge of the annular region; the +other two edges are spokes. + +So each $d_f$ has degree $3$ in $D(T)$: two from interior edges (= spokes +shared with adjacent annular faces) and one leaf. The induced subgraph +on $\{d_f : f \in F_{\mathrm{ann}}\}$ is $2$-regular; together with the +connectedness of the annular region this forces it to be a single +cycle. By Remark~\ref{rem:tire-counts}, the cycle has length $n + m$, +and there are also $n + m$ leaves attached one-per-cycle-vertex. +\end{proof} + \begin{proposition}[Source-side simple-cycle property] \label{prop:no-level-d-pinch} Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and