dual_decomposition: scaffold paper + fill in minimal-counterexample step
New paper "Dual Decomposition of Minimal Counterexamples" outlining a six-step cut-and-recombine attack on the 4CT via the dual cubic graph: minimal counterexample -> dualise -> minimum (cyclic) edge cut -> cap to cubic -> 3-edge-colour the pieces -> reconnect. Strategy section flags steps 1-5 as standard machinery and step 6 (recombination) as the crux. Step 1 written in full: reduction to triangulations, definition of the minimal counterexample, minimum-degree >= 5 (degree <=3 and degree-4 Kempe cases), and no separating triangle => 4-connected. 5-connectivity stated as Birkhoff's separating-4-cycle reduction (attributed, not re-derived). Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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%% filename: amsart-template.tex
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%% American Mathematical Society
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%% AMS-LaTeX v.2 template for use with amsart
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%% ====================================================================
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\documentclass{amsart}
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\usepackage{amssymb}
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\usepackage{graphicx}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{corollary}[theorem]{Corollary}
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\theoremstyle{definition}
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\newtheorem{example}[theorem]{Example}
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\theoremstyle{remark}
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\newtheorem{remark}[theorem]{Remark}
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\numberwithin{equation}{section}
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\begin{document}
|
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|
||||
\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}
|
||||
Reference in New Issue
Block a user