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+\citation{bauerfeld-medial-tire} +\citation{bauerfeld-medial-tire} +\newlabel{lem:kempe-conservation}{{3.2}{3}} +\newlabel{def:kempe-balanced}{{3.3}{3}} +\newlabel{rem:kempe-balance-necessary}{{3.4}{3}} +\bibcite{bauerfeld-medial-tire}{1} +\bibcite{bauerfeld-nested-tire-decompositions}{2} +\bibcite{tait-original}{3} +\newlabel{tocindent-1}{0pt} +\newlabel{tocindent0}{12.7778pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{0pt} +\newlabel{tocindent3}{0pt} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{}\protected@file@percent } +\gdef \@abspage@last{6} diff --git a/papers/medial_pigeonhole_programme/paper.fdb_latexmk b/papers/medial_pigeonhole_programme/paper.fdb_latexmk new file mode 100644 index 0000000..f3621ce --- /dev/null +++ b/papers/medial_pigeonhole_programme/paper.fdb_latexmk @@ -0,0 +1,135 @@ +# Fdb version 3 +["pdflatex"] 1781485473 "paper.tex" "paper.pdf" "paper" 1781485473 + 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-0,0 +1,442 @@ +%% filename: amsart-template.tex +%% American Mathematical Society +%% AMS-LaTeX v.2 template for use with amsart +%% ==================================================================== + +\documentclass{amsart} + +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{tikz} +\usetikzlibrary{backgrounds} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{conjecture}[theorem]{Conjecture} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\newtheorem{xca}[theorem]{Exercise} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\numberwithin{equation}{section} + +\begin{document} + +\title{The Medial Pigeonhole Programme} + +% author one information +\author{Eric Bauerfeld} +\address{} +\curraddr{} +\email{} +\thanks{} + +\subjclass[2010]{Primary } + +\keywords{plane graph, triangulation, medial graph, tire graph, Tait coloring, Kempe chain, Four Colour Theorem} + +\date{} + +\dedicatory{} + +\begin{abstract} +Building on the medial tire decomposition of a plane triangulation, we +formulate a pigeonhole programme for the Four Colour Theorem in medial +terms. Each tire carries a boundary-state restriction relation, and a +proper vertex $3$-colouring of the full medial graph is a compatible +selection of these boundary states across the tire tree. We state a +chain-pigeonhole conjecture asserting that the restriction relations +cannot remain mutually disjoint along every branch, and we refine the +boundary states by recording how two-colour Kempe cycles are routed +through each annular tire region. This yields a Kempe-enhanced +restriction relation and a notion of Kempe-compatible gluing along level +cycles. +\end{abstract} + +\maketitle + +\section{Introduction} + +This paper continues the medial tire programme begun +in~\cite{bauerfeld-medial-tire}. We use freely the terminology and +notation introduced there. For a plane triangulation $G$ with fixed +embedding, $M(G)$ denotes the full medial graph, and the tire-tree +decomposition $\mathcal{T}(G,S)$ at a level source $S$ +of~\cite{bauerfeld-nested-tire-decompositions} induces a decomposition +of $M(G)$ into full medial tire graphs $\mathsf{M}(T)$, one for each +tread $T$, glued along their boundary medial vertex sets +$\partial_{\mathrm{out}}\mathsf{M}(T)$ and +$\partial_{\mathrm{in}}\mathsf{M}(T)$. We also use the annular medial +cycle $A(T)$, its up and down teeth and their apexes, the bites and the +auxiliary plane graph $B(T)$, and the medial tire restriction relation +$R_T$ of~\cite{bauerfeld-medial-tire}. + +By the Tait--medial correspondence of~\cite{bauerfeld-medial-tire}, +proper vertex $3$-colourings of $M(G)$ are in natural bijection with +proper $3$-edge-colourings of the cubic planar dual $G^*$. Thus the +Four Colour Theorem is the assertion that the full medial graph of every +plane triangulation is properly vertex $3$-colourable, and the medial +tire decomposition turns this into a question about how local boundary +colourings compose across the tire tree. + +\section{A medial pigeonhole programme} + +The restriction relation $R_T$ records exactly the local information +needed to pass a medial $3$-colouring through a tire. In a nested +chain +\[ + T_0 \supset T_1 \supset \cdots \supset T_k, +\] +the outer boundary state of $T_{i+1}$ must match an inner boundary +state allowed by $R_{T_i}$. Thus a proof of the Four Colour Theorem in +this framework would follow from a structural reason that these +restriction sets cannot remain mutually disjoint along every branch of +the tire tree. + +\begin{definition}[Medial boundary state] +\label{def:medial-boundary-state} +A \emph{medial boundary state} on a boundary set +$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the +subgraph induced by that boundary set, considered up to permutation of +the three colours and the dihedral symmetries of the boundary walk +when that boundary is a cycle. +\end{definition} + +\begin{conjecture}[Medial chain-pigeonhole principle] +\label{conj:medial-chain-pigeonhole} +There is a function $N(k)$ such that the following holds. Let +$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of +tire treads whose relevant boundary medial walks have length at most +$k$. Then two adjacent restriction relations in the chain have +compatible medial boundary states after colour permutation and boundary +symmetry. Equivalently, the chain contains a local gluing step that +cannot be obstructed by disjoint proper vertex $3$-colouring +restrictions. +\end{conjecture} + +\begin{conjecture}[Medial tire route to the Four Colour Theorem] +\label{conj:medial-route-fct} +For every plane triangulation $G$ and every level source $S$, the +restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a +compatible selection of boundary states across the tire tree. Hence +$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly +$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable. +\end{conjecture} + +\begin{remark} +Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to +the Four Colour Theorem when combined with Tait's correspondence. The +point of the formulation is not to weaken the target theorem, but to +move the obstruction into finite boundary-state restrictions carried by +annular medial tire pieces. +\end{remark} + +\section{Kempe-cycle conservation across medial tires} + +We now record an additional structure carried by proper +$3$-colourings of medial graphs. This structure will be useful for +describing how colourings glue across level cycles. + +Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph. +Let +\[ + \varphi:V(M)\to\{1,2,3\} +\] +be a proper $3$-colouring of $M$. For a two-element colour set +$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$ +induced by the vertices of colours $a$ and $b$. + +Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of +$M_P$ has degree $2$ in $M_P$. Hence every component of $M_P$ is a +cycle. We call these components the $P$-Kempe cycles of $\varphi$. + +\begin{lemma}[Kempe chains are cycles] +\label{lem:kempe-cycles} +Let $G$ be a plane triangulation, let $M=M(G)$, and let +$\varphi$ be a proper $3$-colouring of $M$. For each +$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle. +\end{lemma} + +\begin{proof} +Let $v\in V(M_P)$. In the medial graph $M$, the vertex $v$ has degree +$4$. Since $\varphi$ is a proper $3$-colouring, none of the neighbours +of $v$ has colour $\varphi(v)$. Thus all four neighbours of $v$ have +one of the two colours different from $\varphi(v)$. + +In the medial graph of a plane triangulation, the neighbours of a +medial vertex occur in two opposite pairs corresponding to the two +faces incident with the corresponding edge of $G$. Around each such +triangular face, the three medial vertices receive all three colours. +Consequently, at $v$ there are exactly two neighbours of each colour +different from $\varphi(v)$. It follows that, in the subgraph induced +by any two colours $P$, every vertex has degree $2$. Hence each +component of $M_P$ is a cycle. +\end{proof} + +Let $T$ be a medial tire region. We regard $T$ as an annular transition +region whose boundary consists of one outer level cycle and finitely +many inner level cycles: +\[ + \partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m. +\] +Here $C_0$ is the outer level cycle of $T$, and the cycles +$C_1,\ldots,C_m$ are the inner level cycles. Each inner level cycle +$C_i$ is also the outer level cycle of the corresponding child region +in the tire tree. + +The following lemma is the basic conservation principle. + +\begin{lemma}[Kempe-cycle conservation across level cycles] +\label{lem:kempe-conservation} +Let $C$ be a level cycle of $M$ separating a parent side from a child +side. Let $K$ be a $P$-Kempe cycle for some +$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$. Then $K$ cannot enter the child side +of $C$ without also leaving it. + +Equivalently, the incidences of $K$ with $C$ are paired by the +components of $K$ lying on the child side of $C$, and also paired by the +components of $K$ lying on the parent side of $C$. +\end{lemma} + +\begin{proof} +By the preceding lemma, $K$ is a cycle. The level cycle $C$ separates +the sphere into two closed regions, which we call the parent side and +the child side. Consider the intersection of $K$ with one of these +regions. Since $K$ is a cycle, no component of this intersection can +have exactly one boundary endpoint on $C$. Each component is either +closed within the region, or is a path with two boundary endpoints on +$C$. 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 +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 +$\mathsf{M}(T)$. 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 +$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe +chains} of $\varphi$. Every vertex of colour $a$ or $b$ lies on exactly +one $P$-Kempe chain. + +A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior +face of $B(T)$ that is not a tooth---namely the root face or a bite +inner-gap face in the sense of~\cite{bauerfeld-medial-tire}. The +\emph{tooth apexes incident to} a valid face $F$ are: +\begin{itemize} + \item the up-tooth apexes (\cite{bauerfeld-medial-tire}), when + $F$ is the outer face; + \item the singleton down-tooth apexes whose annular edge lies on $F$, + when $F$ is interior---the apex on annular edge $m$ being incident to + the innermost bite $(i,j)$ with $i -Output written on paper.pdf (11 pages, 277538 bytes). + + + +Output written on paper.pdf (6 pages, 250239 bytes). PDF statistics: - 138 PDF objects out of 1000 (max. 8388607) - 86 compressed objects within 1 object stream + 121 PDF objects out of 1000 (max. 8388607) + 74 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 13 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf index 61e577f..90794b0 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/paper.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex b/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex index 1e50816..6b05235 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex +++ b/papers/medial_tire_decompositions_of_plane_triangulations/paper.tex @@ -53,9 +53,12 @@ isomorphic to the medial graph of the planar dual $G^*$, and proper $3$-vertex-colourings of $M(G)$ are equivalent to proper $3$-edge-colourings of the cubic dual. Thus Tait's reformulation of the Four Colour Theorem may be studied through proper vertex -$3$-colourings of medial subgraphs. We define medial tire pieces, -their boundary-state restriction relations, and a chain-pigeonhole -conjecture for compatible medial boundary states across the tire tree. +$3$-colourings of medial subgraphs. We define medial tire pieces and +their boundary-state restriction relations, and show that a proper +vertex $3$-colouring of $M(G)$ amounts to a compatible selection of +these boundary states across the tire tree. The resulting +pigeonhole programme for the Four Colour Theorem is developed +in~\cite{bauerfeld-medial-pigeonhole}. \end{abstract} \maketitle @@ -656,340 +659,6 @@ properness is already enforced by one of the local colourings. Hence $\varphi$ is a proper vertex $3$-colouring of $M(G)$. \end{proof} -\section{A medial pigeonhole programme} - -The restriction relation $R_T$ records exactly the local information -needed to pass a medial $3$-colouring through a tire. In a nested -chain -\[ - T_0 \supset T_1 \supset \cdots \supset T_k, -\] -the outer boundary state of $T_{i+1}$ must match an inner boundary -state allowed by $R_{T_i}$. Thus a proof of the Four Colour Theorem in -this framework would follow from a structural reason that these -restriction sets cannot remain mutually disjoint along every branch of -the tire tree. - -\begin{definition}[Medial boundary state] -\label{def:medial-boundary-state} -A \emph{medial boundary state} on a boundary set -$\partial\mathsf{M}(T)$ is a proper vertex $3$-colouring of the -subgraph induced by that boundary set, considered up to permutation of -the three colours and the dihedral symmetries of the boundary walk -when that boundary is a cycle. -\end{definition} - -\begin{conjecture}[Medial chain-pigeonhole principle] -\label{conj:medial-chain-pigeonhole} -There is a function $N(k)$ such that the following holds. Let -$T_0 \supset T_1 \supset \cdots \supset T_{N(k)}$ be a nested chain of -tire treads whose relevant boundary medial walks have length at most -$k$. Then two adjacent restriction relations in the chain have -compatible medial boundary states after colour permutation and boundary -symmetry. Equivalently, the chain contains a local gluing step that -cannot be obstructed by disjoint proper vertex $3$-colouring -restrictions. -\end{conjecture} - -\begin{conjecture}[Medial tire route to the Four Colour Theorem] -\label{conj:medial-route-fct} -For every plane triangulation $G$ and every level source $S$, the -restriction relations $\{R_T : T \in V(\mathcal{T}(G,S))\}$ admit a -compatible selection of boundary states across the tire tree. Hence -$M(G)$ is properly vertex $3$-colourable, $G^*$ is properly -$3$-edge-colourable, and $G$ is properly $4$-vertex-colourable. -\end{conjecture} - -\begin{remark} -Conjecture~\ref{conj:medial-route-fct} is equivalent in strength to -the Four Colour Theorem when combined with Tait's correspondence. The -point of the formulation is not to weaken the target theorem, but to -move the obstruction into finite boundary-state restrictions carried by -annular medial tire pieces. -\end{remark} - -\subsection{Kempe-cycle conservation across medial tires} - -We now record an additional structure carried by proper -$3$-colourings of medial graphs. This structure will be useful for -describing how colourings glue across level cycles. - -Let $G$ be a plane triangulation and let $M=M(G)$ be its medial graph. -Let -\[ - \varphi:V(M)\to\{1,2,3\} -\] -be a proper $3$-colouring of $M$. For a two-element colour set -$P=\{a,b\}\subseteq\{1,2,3\}$, let $M_P$ denote the subgraph of $M$ -induced by the vertices of colours $a$ and $b$. - -Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of -$M_P$ has degree $2$ in $M_P$. Hence every component of $M_P$ is a -cycle. We call these components the $P$-Kempe cycles of $\varphi$. - -\begin{lemma}[Kempe chains are cycles] -\label{lem:kempe-cycles} -Let $G$ be a plane triangulation, let $M=M(G)$, and let -$\varphi$ be a proper $3$-colouring of $M$. For each -$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle. -\end{lemma} - -\begin{proof} -Let $v\in V(M_P)$. In the medial graph $M$, the vertex $v$ has degree -$4$. Since $\varphi$ is a proper $3$-colouring, none of the neighbours -of $v$ has colour $\varphi(v)$. Thus all four neighbours of $v$ have -one of the two colours different from $\varphi(v)$. - -In the medial graph of a plane triangulation, the neighbours of a -medial vertex occur in two opposite pairs corresponding to the two -faces incident with the corresponding edge of $G$. Around each such -triangular face, the three medial vertices receive all three colours. -Consequently, at $v$ there are exactly two neighbours of each colour -different from $\varphi(v)$. It follows that, in the subgraph induced -by any two colours $P$, every vertex has degree $2$. Hence each -component of $M_P$ is a cycle. -\end{proof} - -Let $T$ be a medial tire region. We regard $T$ as an annular transition -region whose boundary consists of one outer level cycle and finitely -many inner level cycles: -\[ - \partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m. -\] -Here $C_0$ is the outer level cycle of $T$, and the cycles -$C_1,\ldots,C_m$ are the inner level cycles. Each inner level cycle -$C_i$ is also the outer level cycle of the corresponding child region -in the tire tree. - -The following lemma is the basic conservation principle. - -\begin{lemma}[Kempe-cycle conservation across level cycles] -\label{lem:kempe-conservation} -Let $C$ be a level cycle of $M$ separating a parent side from a child -side. Let $K$ be a $P$-Kempe cycle for some -$P\in\{\{1,2\},\{2,3\},\{3,1\}\}$. Then $K$ cannot enter the child side -of $C$ without also leaving it. - -Equivalently, the incidences of $K$ with $C$ are paired by the -components of $K$ lying on the child side of $C$, and also paired by the -components of $K$ lying on the parent side of $C$. -\end{lemma} - -\begin{proof} -By the preceding lemma, $K$ is a cycle. The level cycle $C$ separates -the sphere into two closed regions, which we call the parent side and -the child side. Consider the intersection of $K$ with one of these -regions. Since $K$ is a cycle, no component of this intersection can -have exactly one boundary endpoint on $C$. Each component is either -closed within the region, or is a path with two boundary endpoints on -$C$. 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 -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 -$\mathsf{M}(T)$. 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 -$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe -chains} of $\varphi$. Every vertex of colour $a$ or $b$ lies on exactly -one $P$-Kempe chain. - -A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior -face of $B(T)$ that is not a tooth---namely the root face or a bite -inner-gap face of Remark~\ref{rem:bite-face-count}. The \emph{tooth -apexes incident to} a valid face $F$ are: -\begin{itemize} - \item the up-tooth apexes (Definition~\ref{def:annular-teeth}), when - $F$ is the outer face; - \item the singleton down-tooth apexes whose annular edge lies on $F$, - when $F$ is interior---the apex on annular edge $m$ being incident to - the innermost bite $(i,j)$ with $i