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b/papers/dual_decomposition_minimal_counterexamples/paper.tex @@ -0,0 +1,246 @@ +%% filename: amsart-template.tex +%% American Mathematical Society +%% AMS-LaTeX v.2 template for use with amsart +%% ==================================================================== + +\documentclass{amsart} + +\usepackage{amssymb} +\usepackage{graphicx} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{proposition}[theorem]{Proposition} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\numberwithin{equation}{section} + +\begin{document} + +\title{Dual Decomposition of Minimal Counterexamples} + +% author one information +\author{Eric Bauerfeld} +\address{} +\curraddr{} +\email{} +\thanks{} + +\subjclass[2010]{Primary } + +\keywords{four colour theorem, plane triangulation, dual graph, cubic planar +graph, edge connectivity, cyclic edge cut, Tait colouring, $3$-edge-colouring} + +\date{} + +\dedicatory{} + +\begin{abstract} +% TODO: abstract. +\end{abstract} + +\maketitle + +\section{Introduction} + +% TODO: introduction. + +By Tait's theorem, a plane triangulation $G$ is properly $4$-vertex-colourable if +and only if its dual cubic graph $G'$ is properly $3$-edge-colourable. We propose +to obstruct a minimal counterexample to the Four Colour Theorem by decomposing its +dual along a minimum edge cut, $3$-edge-colouring the two resulting pieces, and +recombining the colourings. + +\section{Strategy} + +% Outline of the intended argument. Each step below is to be made precise and +% proved in its own section; nothing here is yet established. + +The argument proceeds in six steps. + +\begin{enumerate} + \item \textbf{Minimal counterexample.} Assume a smallest maximal planar graph + $G$ admitting no proper $4$-colouring. Standard reductions let us take $G$ + to be $5$-connected, so its dual $G'$ is a cyclically $5$-edge-connected + cubic plane graph. + + \item \textbf{Dualise.} Pass to the dual cubic plane graph $G'$, where, by + Tait, $4$-colourability of $G$ is equivalent to $3$-edge-colourability of + $G'$. + + \item \textbf{Minimum edge cut.} Take a smallest edge cut of $G'$ separating it + into two components $A$ and $B$. (The size of the smallest cyclic edge cut + equals the cyclic edge connectivity of $G'$; record which cut is used and + why it is minimal.) + + \item \textbf{Cap to cubic.} Add the minimum number of edges (and, if needed, + vertices) to each of $A$ and $B$ to absorb the severed cut edges, yielding + two cubic plane graphs $A^{+}$ and $B^{+}$, each smaller than $G'$. + + \item \textbf{Colour the pieces.} Since $A^{+}$ and $B^{+}$ are smaller than the + minimal counterexample's dual, each admits a proper $3$-edge-colouring. + + \item \textbf{Reconnect.} Prove that the two $3$-edge-colourings can be made + compatible across the cut --- after a colour permutation and Kempe-type + adjustments --- so that they glue to a proper $3$-edge-colouring of $G'$, + contradicting the assumption that $G$ is a counterexample. +\end{enumerate} + +\begin{remark} +Steps (1)--(5) are assembled from standard machinery (minimality, Tait duality, +edge-cut/girth duality, and the inductive existence of colourings on smaller cubic +graphs). The load-bearing claim is step (6): the compatibility of the two +boundary colourings across the cut. The parity of colours along an edge cut in a +$3$-edge-colouring is constrained, and the goal is to show these constraints can +always be satisfied simultaneously on both sides. +\end{remark} + +\section{Step 1: The minimal counterexample} + +Throughout, a \emph{triangulation} is a simple plane graph, with a fixed +embedding, in which every face --- including the outer face --- is bounded by a +triangle. We first reduce to triangulations, then record the degree and +connectivity properties a smallest counterexample must have. + +\begin{lemma}[Reduction to triangulations] +\label{lem:triangulate} +If every triangulation is properly $4$-vertex-colourable, then so is every plane +graph. +\end{lemma} + +\begin{proof} +Let $H$ be a plane graph. Add edges to $H$, maintaining planarity, until no +further edge can be added; the result is a triangulation $H^{+}$ on the same +vertex set with $E(H) \subseteq E(H^{+})$. A proper $4$-colouring of $H^{+}$ +restricts to a proper $4$-colouring of $H$, since every edge of $H$ is an edge of +$H^{+}$. +\end{proof} + +By Lemma~\ref{lem:triangulate}, if the Four Colour Theorem fails then it fails for +some triangulation. We may therefore make the following assumption. + +\begin{definition}[Minimal counterexample] +\label{def:minimal} +Let $G$ be a triangulation on the fewest vertices that admits no proper +$4$-vertex-colouring. We call $G$ a \emph{minimal counterexample}. By minimality, +every triangulation on fewer than $|V(G)|$ vertices is properly +$4$-colourable. +\end{definition} + +\begin{remark} +Since every triangulation on at most four vertices is properly $4$-colourable +(the largest being $K_4$), a minimal counterexample has $|V(G)| \ge 5$; the degree +bound below sharpens this to $|V(G)| \ge 12$. +\end{remark} + +\begin{lemma}[Minimum degree] +\label{lem:mindeg} +A minimal counterexample $G$ has minimum degree $\delta(G) \ge 5$. +\end{lemma} + +\begin{proof} +Suppose some vertex $v$ has $\deg(v) = d \le 4$. + +If $d \le 3$, let $G' = G - v$. Then $G'$ is a plane graph on fewer vertices, so +by Definition~\ref{def:minimal} and Lemma~\ref{lem:triangulate} it has a proper +$4$-colouring. The at most three neighbours of $v$ use at most three colours, so +a fourth colour is free for $v$, extending the colouring to $G$ --- a +contradiction. + +If $d = 4$, again $4$-colour $G - v$. If the four neighbours of $v$ use at most +three colours we extend as before, so assume they receive all four colours; let +$v_1, v_2, v_3, v_4$ be the neighbours in cyclic order around $v$, coloured +$1,2,3,4$. Consider the subgraph induced by the colour classes $1$ and $3$, and +let $K$ be its connected component containing $v_1$. If $v_3 \notin K$, swap +colours $1$ and $3$ on $K$; now no neighbour of $v$ is coloured $1$, freeing it +for $v$. If $v_3 \in K$, then a $1$--$3$ Kempe chain joins $v_1$ to $v_3$, and +this chain together with $v$ encloses exactly one of $v_2, v_4$; hence the +$2$--$4$ component containing $v_2$ cannot also reach $v_4$, and swapping colours +$2$ and $4$ on it frees colour $2$ for $v$. Either way the colouring extends to +$G$, a contradiction. + +Hence $\delta(G) \ge 5$. +\end{proof} + +For the connectivity reduction we recall that, in a triangulation, the relevant +small separators are short cycles. + +\begin{definition}[Separating cycle] +A cycle $C$ in a triangulation $G$ is \emph{separating} if it is not a face +boundary; equivalently, both the open interior and the open exterior of $C$ (with +respect to the embedding) contain at least one vertex of $G$. +\end{definition} + +\begin{lemma}[No separating triangle] +\label{lem:no-sep-tri} +A minimal counterexample $G$ has no separating triangle; consequently $G$ is +$4$-connected. +\end{lemma} + +\begin{proof} +Suppose $T = xyz$ is a separating triangle. It splits $G$ into the triangulation +$G_{\mathrm{in}}$ induced by $T$ and the vertices interior to $T$, and the +triangulation $G_{\mathrm{out}}$ induced by $T$ and the vertices exterior to $T$; +in each, $T$ bounds a face. Both are triangulations on fewer vertices than $G$, +so by minimality each has a proper $4$-colouring. Permuting colours in +$G_{\mathrm{out}}$, we may assume the two colourings agree on $\{x,y,z\}$ (the +three corners receive distinct colours in each, so a colour permutation aligns +them). The two colourings then glue to a proper $4$-colouring of $G$, a +contradiction. A triangulation with no separating triangle and $\delta \ge 5$ is +$4$-connected, since a minimal vertex cut of size $\le 3$ in a triangulation is +the vertex set of a separating cycle of that length, and a $3$-cut would give a +separating triangle. +\end{proof} + +\begin{proposition}[$5$-connectivity] +\label{prop:5conn} +A minimal counterexample $G$ may be taken to be $5$-connected; equivalently, $G$ +has no separating triangle and no separating $4$-cycle. +\end{proposition} + +\begin{proof}[Discussion] +By Lemma~\ref{lem:no-sep-tri} it remains to eliminate separating $4$-cycles. This +is the classical reducibility step of Birkhoff: a shortest separating $4$-cycle +$C = w x y z$ bounds inner and outer triangulations; one $4$-colours both by +minimality and reconciles their colourings on $C$ using Kempe-chain exchanges, +the only obstruction being the two ways the four corners can be coloured with +three or four colours. Carrying out the exchanges shows the colourings can always +be aligned, so $G$ has no separating $4$-cycle. As with +Lemma~\ref{lem:no-sep-tri}, the absence of separating $3$- and $4$-cycles in a +triangulation with $\delta \ge 5$ is equivalent to $5$-connectivity. We take this +reduction as standard and refer to the literature on the Four Colour Theorem for +the detailed Kempe-chain bookkeeping. +\end{proof} + +The output of this step --- a $5$-connected triangulation $G$ with $\delta(G) \ge +5$ and no separating triangle or $4$-cycle --- is exactly the hypothesis the dual +construction of Step~2 consumes. + +\section{Step 2: The dual cubic graph} + +% TODO. + +\section{Step 3: The minimum edge cut} + +% TODO. + +\section{Step 4: Capping to cubic planar graphs} + +% TODO. + +\section{Step 5: Colouring the pieces} + +% TODO. + +\section{Step 6: Reconnecting the colourings} + +% TODO. (The crux.) + +\end{document}