diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 9d8e8b2..450c2a8 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -1,11 +1,12 @@ \relax +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } +\newlabel{def:dual-depth}{{1.4}{1}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } -\newlabel{def:dual-depth}{{1.4}{1}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent } \newlabel{fig:dual-depth}{{1}{2}} -\gdef \@abspage@last{2} +\newlabel{def:tire-graph}{{1.5}{2}} +\gdef \@abspage@last{3} diff --git a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk index 2508ddb..004a41f 100644 --- a/papers/coloring_nested_tire_graphs/paper.fdb_latexmk +++ b/papers/coloring_nested_tire_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1779482777 "paper.tex" "paper.pdf" "paper" 1779482778 +["pdflatex"] 1779733650 "paper.tex" "paper.pdf" "paper" 1779733650 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -56,8 +56,8 @@ "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" "fig_dual_depth.png" 1779482522 255786 cb48aab5aa40fc161d13a75df0544511 "" - "paper.aux" 1779482778 941 aea7c9fac695aa780643110af03da54c "pdflatex" - "paper.tex" 1779482773 4064 135510a3466a93155ee81a8d88e3ce4c "" + "paper.aux" 1779733650 977 9ae602f68da02f011f6275321bae63a1 "pdflatex" + "paper.tex" 1779733644 5855 ae8bddac7e77a789fdbf38b2af99908f "" (generated) "paper.aux" "paper.log" diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 5b263d5..9c81e28 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) 22 MAY 2026 16:46 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 14:27 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -201,32 +201,32 @@ Package pdftex.def Info: fig_dual_depth.png used on input line 106. 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PDF statistics: - 74 PDF objects out of 1000 (max. 8388607) - 43 compressed objects within 1 object stream + 77 PDF objects out of 1000 (max. 8388607) + 45 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 6 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 334cb6f..5a7f442 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 d6f9787..3875fca 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -25,7 +25,7 @@ \begin{document} -\title{Nested Level Duals} +\title{Coloring Nested Tire Graphs} % author one information \author{Eric Bauerfeld} @@ -114,8 +114,41 @@ vertex.} \label{fig:dual-depth} \end{figure} -\end{document} +\begin{definition}[Tire graph] +\label{def:tire-graph} +Let $C_{\mathrm{out}}$ be a simple cycle of length $m \geq 3$, and let +$O$ be an outerplanar graph whose outer-face boundary $C_{\mathrm{in}}$ +is a simple cycle of length $k \geq 3$, with $V(C_{\mathrm{out}}) \cap +V(O) = \emptyset$. A \emph{tire graph} on $(C_{\mathrm{out}}, O)$ is a +plane graph $T$ with +\[ + V(T) = V(C_{\mathrm{out}}) \cup V(O), + \qquad + E(T) = E(C_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}}, +\] +where $E_{\mathrm{ann}}$ is a set of edges --- the \emph{annular edges} +--- such that, in the plane embedding of $T$, the closed annulus with +outer boundary $C_{\mathrm{out}}$ and inner boundary $C_{\mathrm{in}}$ +is partitioned into triangular faces. Equivalently, the bounded faces +of $T$ that are not faces of $O$ are all triangles, and together they +tile the annular region between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$. -% NOTE (2026-05-22): This paper is being shelved in favour of an alternative -% approach. The nested-level-duals framing is preserved here for reference but -% is not being actively developed. +We call $C_{\mathrm{out}}$ the \emph{outer cycle}, $O$ the \emph{inner +outerplanar graph}, and $C_{\mathrm{in}}$ the \emph{inner cycle} of +$T$. When $O = C_{\mathrm{in}}$ (the inner outerplanar graph has no +chords), $T$ is a tire graph \emph{with empty inner}; in general $O$ +contributes only chords inside the disk bounded by $C_{\mathrm{in}}$ +and does not interact with $E_{\mathrm{ann}}$. +\end{definition} + +\begin{remark} +A tire graph on $(C_{\mathrm{out}}, O)$ has $|V(C_{\mathrm{out}})| + +|V(O)| = m + k$ vertices, exactly $m + k$ annular triangles +in the annulus between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$ (by +Euler's formula on the annulus), and exactly $m + k$ annular edges +in $E_{\mathrm{ann}}$, of which the $m + k$ triangles share their +three edges with the boundaries $E(C_{\mathrm{out}}) \cup +E(C_{\mathrm{in}})$ and with each other. +\end{remark} + +\end{document}