From 2a563228415a63154050333f5f29984f88cba34f Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 15 Jun 2026 16:14:50 -0400 Subject: [PATCH] Define simple and compound medial tires --- .../paper.aux | 28 +++++----- .../paper.fdb_latexmk | 6 +-- .../paper.log | 13 +++-- .../paper.pdf | Bin 277538 -> 278916 bytes .../paper.tex | 51 ++++++++++++------ 5 files changed, 61 insertions(+), 37 deletions(-) diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux b/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux index 08b9e87..ecc1017 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux +++ b/papers/medial_tire_decompositions_of_plane_triangulations/paper.aux @@ -12,33 +12,33 @@ \newlabel{def:full-medial-tire}{{3.1}{2}} \newlabel{thm:annular-medial-colour-bound}{{3.3}{3}} \newlabel{def:annular-teeth}{{3.4}{3}} -\newlabel{rem:teeth-sharing}{{3.5}{3}} -\newlabel{rem:up-teeth-count}{{3.6}{3}} +\citation{bauerfeld-nested-tire-decompositions} +\citation{bauerfeld-nested-tire-decompositions} +\newlabel{rem:teeth-sharing}{{3.5}{4}} +\newlabel{rem:up-teeth-count}{{3.6}{4}} \newlabel{def:bite}{{3.7}{4}} \newlabel{rem:bite-face-count}{{3.8}{4}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A full medial tire graph $\mathsf {M}(T)$ illustrating the tooth terminology. The thick cycle is the annular medial cycle $A(T)$, whose black vertices are the annular medial vertices. Each edge of $A(T)$ carries one tooth: up teeth (blue apexes, outer-boundary medial vertices) point into the outer region, and down teeth (red apexes, inner-boundary medial vertices) point into the inner region. The two down teeth meeting at the central shared apex (larger red vertex) form a bite; that shared apex splits the inner region into two faces, one with four down teeth on its boundary and one with none.}}{4}{}\protected@file@percent } -\newlabel{fig:medial-teeth-example}{{1}{4}} \newlabel{def:boundary-medial-vertices}{{3.9}{4}} -\citation{bauerfeld-nested-tire-decompositions} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Three six-face full medial tire graphs found by the boundary-state restriction search. Black vertices are annular medial vertices; blue vertices are outer boundary medial vertices and red vertices are inner boundary medial vertices. The word below each diagram records the outer/inner type of the six annular faces in cyclic order. Boundary states are identified only up to colour permutation, not by rotation or reflection of the boundary order.}}{5}{}\protected@file@percent } +\newlabel{def:medial-restriction-relation}{{3.10}{4}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{4}{}\protected@file@percent } +\newlabel{cor:medial-tire-decomposition}{{4.1}{4}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A simple medial tire graph $\mathsf {M}(T)$ illustrating the tooth terminology. The thick cycle is the annular medial cycle $A(T)$, whose black vertices are the annular medial vertices. Each edge of $A(T)$ carries one tooth: up teeth (blue apexes, outer-boundary medial vertices) point into the outer region, and down teeth (red apexes, inner-boundary medial vertices) point into the inner region. The two down teeth meeting at the central shared apex (larger red vertex) form a bite; that shared apex splits the inner region into two faces, one with four down teeth on its boundary and one with none.}}{5}{}\protected@file@percent } +\newlabel{fig:medial-teeth-example}{{1}{5}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Three six-face simple medial tire graphs found by the boundary-state restriction search. Black vertices are annular medial vertices; blue vertices are outer boundary medial vertices and red vertices are inner boundary medial vertices. The word below each diagram records the outer/inner type of the six annular faces in cyclic order. Boundary states are identified only up to colour permutation, not by rotation or reflection of the boundary order.}}{5}{}\protected@file@percent } \newlabel{fig:medial-restriction-worst-cases}{{2}{5}} -\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A proper vertex $3$-colouring of the full medial graph of the first seven-vertex counterexample found by the experiment. The medial vertex labelled $ij$ corresponds to the edge $(i,j)$ of the triangulation. For the vertex-source decomposition at source $1$, the highlighted annular medial cycle has colour counts $(2,2,2)$, so it is not coloured with two colours except at at most one vertex.}}{5}{}\protected@file@percent } -\newlabel{fig:medial-annular-cycle-counterexample}{{3}{5}} -\newlabel{def:medial-restriction-relation}{{3.10}{5}} -\citation{bauerfeld-nested-tire-decompositions} -\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{6}{}\protected@file@percent } -\newlabel{cor:medial-tire-decomposition}{{4.1}{6}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A proper vertex $3$-colouring of the full medial graph of the first seven-vertex counterexample found by the experiment. The medial vertex labelled $ij$ corresponds to the edge $(i,j)$ of the triangulation. For the vertex-source decomposition at source $1$, the highlighted annular medial cycle has colour counts $(2,2,2)$, so it is not coloured with two colours except at at most one vertex.}}{6}{}\protected@file@percent } +\newlabel{fig:medial-annular-cycle-counterexample}{{3}{6}} \newlabel{def:compatible-family}{{4.2}{6}} \newlabel{prop:gluing-criterion}{{4.3}{6}} \@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{6}{}\protected@file@percent } -\newlabel{def:medial-boundary-state}{{5.1}{6}} +\newlabel{def:medial-boundary-state}{{5.1}{7}} \newlabel{conj:medial-chain-pigeonhole}{{5.2}{7}} \newlabel{conj:medial-route-fct}{{5.3}{7}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Kempe-cycle conservation across medial tires}}{7}{}\protected@file@percent } \newlabel{lem:kempe-cycles}{{5.5}{7}} \newlabel{lem:kempe-conservation}{{5.6}{8}} \newlabel{def:kempe-balanced}{{5.7}{8}} -\newlabel{rem:kempe-balance-necessary}{{5.8}{8}} +\newlabel{rem:kempe-balance-necessary}{{5.8}{9}} \bibcite{bauerfeld-nested-tire-decompositions}{1} \bibcite{tait-original}{2} \newlabel{tocindent-1}{0pt} diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk b/papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk index 09ca62f..3fb5663 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk +++ b/papers/medial_tire_decompositions_of_plane_triangulations/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1781210675 "paper.tex" "paper.pdf" "paper" 1781210676 +["pdflatex"] 1781554446 "paper.tex" "paper.pdf" "paper" 1781554447 "/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 "" @@ -132,8 +132,8 @@ "/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 "" "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" - "paper.aux" 1781210676 4206 870862ca1c6762f39fd7ed9def109a09 "pdflatex" - "paper.tex" 1781210650 40922 403b0b9df57192dbf02362b0b06705c3 "" + "paper.aux" 1781554447 4210 3b62c5dd250f159f0e3d42ba3a3a7308 "pdflatex" + "paper.tex" 1781554440 42216 46cb5902d4210f9324b1231139c3e122 "" (generated) "paper.aux" "paper.log" diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.log b/papers/medial_tire_decompositions_of_plane_triangulations/paper.log index cecc7f4..7b1a511 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/paper.log +++ b/papers/medial_tire_decompositions_of_plane_triangulations/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) 11 JUN 2026 16:44 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 15 JUN 2026 16:14 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -496,7 +496,7 @@ e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] -Overfull \hbox (62.13657pt too wide) in paragraph at lines 363--372 +Overfull \hbox (62.13657pt too wide) in paragraph at lines 371--380 [][] [] @@ -504,17 +504,20 @@ Overfull \hbox (62.13657pt too wide) in paragraph at lines 363--372 LaTeX Warning: `h' float specifier changed to `ht'. +LaTeX Warning: `h' float specifier changed to `ht'. + + LaTeX Warning: `h' float specifier changed to `ht'. 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and compound medial tire graphs] \label{def:full-medial-tire} Let $T$ be a tire tread in the tire tree $\mathcal{T}(G,S)$ supplied -by~\cite{bauerfeld-nested-tire-decompositions}. The \emph{full medial -tire graph} of $T$, denoted $\mathsf{M}(T)$, is the subgraph of -$M(G)$ induced by the medial vertices $m_e$ with $e$ an edge of $G$ -incident to at least one triangular face in the tread $T$. The medial -vertices corresponding to annular edges of $T$ are called -\emph{annular medial vertices}. +by~\cite{bauerfeld-nested-tire-decompositions}. The \emph{medial tire +graph} of $T$, denoted $\mathsf{M}(T)$, is the subgraph of $M(G)$ +induced by the medial vertices $m_e$ with $e$ an edge of $G$ incident +to at least one triangular face in the tread $T$. The medial vertices +corresponding to annular edges of $T$ are called \emph{annular medial +vertices}. + +We call $\mathsf{M}(T)$ a \emph{simple medial tire graph} if its +annular medial vertices induce a single cycle. We call +$\mathsf{M}(T)$ a \emph{compound medial tire graph} if it is associated +to a connected depth component of tread faces but its annular medial +vertices induce more than one cycle. In a compound medial tire graph, +annular teeth are understood cycle-by-cycle, and up-tooth apexes +belonging to different annular cycles may coincide. \end{definition} \begin{remark} -In the ambient-triangulation setting, the full medial tire graph +In the ambient-triangulation setting, the simple medial tire graph $\mathsf{M}(T)$ coincides with the omitted-edge medial tire graph studied in~\cite{bauerfeld-nested-tire-decompositions}. Indeed, the medial edges of $\mathsf{M}(T)$ are contributed by corners of annular @@ -361,7 +369,7 @@ interior. \node[lbl, anchor=east] at (192:1.86) (L0) {region with 0 down teeth}; \draw[lead] (L0.east) -- (180:0.45); \end{tikzpicture} -\caption{A full medial tire graph $\mathsf{M}(T)$ illustrating the tooth +\caption{A simple medial tire graph $\mathsf{M}(T)$ illustrating the tooth terminology. The thick cycle is the annular medial cycle $A(T)$, whose black vertices are the annular medial vertices. Each edge of $A(T)$ carries one tooth: up teeth (blue apexes, outer-boundary medial vertices) @@ -453,7 +461,7 @@ its boundary and one with none.} \node at (0,-2.15) {\scriptsize $\min |R_T(\alpha)|=1$}; \end{scope} \end{tikzpicture} -\caption{Three six-face full medial tire graphs found by the boundary-state +\caption{Three six-face simple medial tire graphs found by the boundary-state restriction search. Black vertices are annular medial vertices; blue vertices are outer boundary medial vertices and red vertices are inner boundary medial vertices. The word below each diagram records the @@ -601,9 +609,11 @@ parent and children. Let $G$ be a plane triangulation with level source $S$. The tire-tree decomposition $\mathcal{T}(G,S)$ of \cite{bauerfeld-nested-tire-decompositions} induces a rooted -decomposition of the full medial graph $M(G)$ into full medial tire -graphs $\{\mathsf{M}(T): T \in V(\mathcal{T}(G,S))\}$, glued along -their boundary medial vertex sets. +decomposition of the full medial graph $M(G)$ into medial tire graphs +$\{\mathsf{M}(T): T \in V(\mathcal{T}(G,S))\}$, glued along their +boundary medial vertex sets. A node of this decomposition may be a +simple medial tire graph or a compound medial tire graph, depending on +whether its annular medial vertices induce one cycle or several. \end{corollary} \begin{proof} @@ -670,6 +680,17 @@ this framework would follow from a structural reason that these restriction sets cannot remain mutually disjoint along every branch of the tire tree. +In this chaining step, the inner side of a parent simple medial tire is +read by its singleton down-tooth apex vertices. If the child side is a +compound medial tire, then the parent's singleton down-tooth apex +vertices are incident to---indeed, are identified with---the up-tooth +apex vertices of the compound medial tire, interpreted cycle-by-cycle +on its annular medial cycles. Equivalently, the primal edges +represented by the parent's singleton down-tooth apexes are exactly the +level-cycle interface edges represented on the child side as up-tooth +apexes. This is the boundary identification along which the medial +boundary states are chained. + \begin{definition}[Medial boundary state] \label{def:medial-boundary-state} A \emph{medial boundary state} on a boundary set @@ -786,12 +807,12 @@ $C$. Thus every entrance through $C$ is paired with an exit through $C$. \end{proof} -We now use these Kempe cycles to single out the colourings of a full +We now use these Kempe cycles to single out the colourings of a simple medial tire graph that respect the annular tooth structure. \begin{definition}[Kempe-balanced colouring] \label{def:kempe-balanced} -Let $\varphi$ be a proper $3$-colouring of the full medial tire graph +Let $\varphi$ be a proper $3$-colouring of the simple medial tire graph $\mathsf{M}(T)$. For a colour pair $P=\{a,b\}$, let $\mathsf{M}(T)_P$ be the subgraph induced by the vertices of colours $a$ and $b$. Since $\mathsf{M}(T)$ need not be $4$-regular, the components of