didericis
41227c6a0f
- Main paper: dual_decomposition_minimal_counterexamples/ -> face_monochromatic_pairs/. Title is now "Face-Monochromatic Pairs and the Four Colour Theorem". - Companion paper: dual_decomposition_iterated_reduction/ -> iterated_reduction_in_reduced_dual/. Title is now "An Iterated Reduction in the Reduced Dual". Its prose and bibliography cite the parent under the new title. - Update one absolute sys.path reference inside check_conj_face_kempe_n15.py that pointed at the old folder. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
834 lines
38 KiB
TeX
834 lines
38 KiB
TeX
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\begin{document}
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\title{Face-Monochromatic Pairs and the Four Colour Theorem}
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% author one information
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\author{Eric Bauerfeld}
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\address{}
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\curraddr{}
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\email{}
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\thanks{}
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\subjclass[2010]{Primary }
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\keywords{four colour theorem, plane triangulation, dual graph, cubic planar
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graph, edge connectivity, cyclic edge cut, Tait colouring, $3$-edge-colouring}
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\date{}
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\dedicatory{}
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\begin{abstract}
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We propose the \emph{face-monochromatic-pair conjecture}, a structural
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property of proper $3$-edge-colourings of cubic plane graphs that, if
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true, implies the Four Colour Theorem. Working in the planar dual $G'$ of
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a hypothetical minimal counterexample $G$ to 4CT, we delete a single
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pentagonal face of $G'$ and rewire its five external vertices around a
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new apex vertex and a chord; the resulting \emph{reduced dual}
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$\widehat{G}'_{v,i}$ is a smaller cubic plane graph whose proper
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$3$-edge-colourings, by the minimality of $G$, are constrained by a
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chord-apex condition and a pair of Kempe-cycle conditions. The
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face-monochromatic-pair conjecture, in its strengthened form, asserts the
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existence in every such colouring of a face $F$ and two non-incident
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same-coloured edges $e_1, e_2 \in \partial F$ whose subdivision-and-%
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bridging produces a $4$-face $f_n$ whose boundary colouring places it
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under the hypothesis of a $4$-face edge-suppression theorem; we use this
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theorem to derive a proper $3$-edge-colouring of $G'$, contradicting
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minimality. We verify the conjecture computationally on all
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chord-apex+Kempe colourings of reduced duals with $|V(G)| \leq 20$
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($142{,}812$ colourings, all pass); the weaker form is verified up to
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$|V(G)| \leq 21$ ($535{,}182$ colourings, all pass).
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\end{abstract}
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\maketitle
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\section{Introduction}
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\label{sec:intro}
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The Four Colour Theorem (4CT) --- that every loopless plane graph admits a
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proper $4$-vertex-colouring --- has, since the late 1970s, only been proved
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by computer-assisted case analysis on a discharging argument over a finite
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set of unavoidable reducible configurations. Appel and Haken's original
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proof~\cite{AH77a, AHK77}, the Robertson--Sanders--Seymour--Thomas
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reworking~\cite{RSST97}, and Gonthier's machine-checked
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version~\cite{Gonthier08} all share that structure.
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This paper takes a different approach: rather than discharge over
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configurations in the triangulation $G$, we work in its planar dual $G'$,
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a cubic plane graph whose proper $3$-edge-colourings correspond by Tait's
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theorem to proper $4$-vertex-colourings of $G$. Assuming $G$ is a minimal
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counterexample to 4CT, we delete a single pentagonal face of $G'$ and
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rewire its five external vertices, obtaining a smaller cubic plane graph
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$\widehat{G}'_{v,i}$ --- the \emph{reduced dual} --- which by minimality
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\emph{is} properly $3$-edge-colourable. Two structural lemmas constrain
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every such colouring: a \emph{chord-apex} condition (Lemma~\ref{lem:chord-apex})
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forcing two named edges to share a colour, and a pair of
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\emph{Kempe-cycle} conditions (Lemma~\ref{lem:kempe-spike}) placing four
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of the rewired edges on common bichromatic Kempe cycles. These constraints
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are the starting point of the present development.
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The main contribution of the paper is the
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\emph{face-monochromatic-pair conjecture}
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(Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}) and
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its strengthening
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(Conjecture~\ref{conj:face-monochromatic-pair-strengthened}), which we
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show together imply the Four Colour Theorem. The supporting ingredients
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are the chord-apex and Kempe-cycle lemmas on reduced-dual colourings, the
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classical operation of \emph{edge suppression} (delete the edge and smooth
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its two degree-$2$ endpoints; equivalently, simple-graph contraction in
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the dual triangulation; recalled in Section~\ref{sec:edge-suppression}),
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and an observation that suppression preserves $3$-edge-colourability when
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applied across a $4$-face whose two opposite boundary edges carry
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different colours (Theorem~\ref{thm:edge-suppression-4face}).
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The strategy is to construct, from a putative minimal counterexample's
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reduced-dual colouring, a $4$-face $f_n$ in a slightly modified graph
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$\widehat{G}'^{+}$ to which the suppression theorem applies; the
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suppression then produces a properly $3$-edge-coloured graph from which a
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$3$-edge-colouring of $G'$ can be recovered, contradicting the
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non-$4$-colourability of $G$. The face-monochromatic-pair conjecture
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asserts the existence of the structural data ($F, e_1, e_2$) needed to
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build $f_n$; the strengthening guarantees that $f_n$'s boundary colouring
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falls under the suppression theorem's hypothesis. Both conjectures have
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been verified computationally on all chord-apex+Kempe colourings of
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reduced duals up to $|V(G)| \leq 20$, with the weaker form going up to
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$|V(G)| \leq 21$.
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\paragraph{Organization.} Section~\ref{sec:minimal} fixes the
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minimal-counterexample framework: $G$ is a triangulation,
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$\delta(G) \geq 5$, and every triangulation on fewer vertices is properly
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$4$-colourable. Section~\ref{sec:reduced-dual} introduces the reduced dual
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$\widehat{G}'_{v,i}$ and proves the chord-apex and Kempe-cycle lemmas.
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Section~\ref{sec:edge-suppression} defines edge suppression and proves
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its $4$-face $3$-edge-colourability theorem.
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Section~\ref{sec:toward-4ct} states the two conjectures, reports the
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empirical verification, and gives the implication to 4CT.
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\paragraph{Companion paper.} An iterated version of the reduced-dual
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construction --- producing a sequence $H_1, H_2, \dots$ of progressively
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smaller cubic plane graphs and tracking an accumulating ``protected'' edge
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set --- is the subject of a companion paper. The present paper does not
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use that iteration.
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\section{The minimal counterexample}
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\label{sec:minimal}
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Throughout, a \emph{triangulation} is a simple plane graph, with a fixed
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embedding, in which every face --- including the outer face --- is bounded by a
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triangle. We first reduce to triangulations, then record the degree properties a
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smallest counterexample must have.
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\begin{lemma}[Reduction to triangulations]
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\label{lem:triangulate}
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If every triangulation is properly $4$-vertex-colourable, then so is every plane
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graph.
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\end{lemma}
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\begin{proof}
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Let $H$ be a plane graph. Add edges to $H$, maintaining planarity, until no
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further edge can be added; the result is a triangulation $H^{+}$ on the same
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vertex set with $E(H) \subseteq E(H^{+})$. A proper $4$-colouring of $H^{+}$
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restricts to a proper $4$-colouring of $H$, since every edge of $H$ is an edge of
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$H^{+}$.
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\end{proof}
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By Lemma~\ref{lem:triangulate}, if the Four Colour Theorem fails then it fails for
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some triangulation. We may therefore make the following assumption.
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\begin{definition}[Minimal counterexample]
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\label{def:minimal}
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Let $G$ be a triangulation on the fewest vertices that admits no proper
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$4$-vertex-colouring. We call $G$ a \emph{minimal counterexample}. By minimality,
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every triangulation on fewer than $|V(G)|$ vertices is properly
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$4$-colourable.
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\end{definition}
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\begin{remark}
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Since every triangulation on at most four vertices is properly $4$-colourable
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(the largest being $K_4$), a minimal counterexample has $|V(G)| \ge 5$; the degree
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bound below sharpens this to $|V(G)| \ge 12$.
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\end{remark}
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\begin{lemma}[Minimum degree]
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\label{lem:mindeg}
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A minimal counterexample $G$ has minimum degree $\delta(G) \ge 5$.
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\end{lemma}
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\begin{proof}
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Suppose some vertex $v$ has $\deg(v) = d \le 4$.
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If $d \le 3$, let $G' = G - v$. Then $G'$ is a plane graph on fewer vertices, so
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by Definition~\ref{def:minimal} and Lemma~\ref{lem:triangulate} it has a proper
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$4$-colouring. The at most three neighbours of $v$ use at most three colours, so
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a fourth colour is free for $v$, extending the colouring to $G$ --- a
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contradiction.
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If $d = 4$, again $4$-colour $G - v$. If the four neighbours of $v$ use at most
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three colours we extend as before, so assume they receive all four colours; let
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$v_1, v_2, v_3, v_4$ be the neighbours in cyclic order around $v$, coloured
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$1,2,3,4$. Consider the subgraph induced by the colour classes $1$ and $3$, and
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let $K$ be its connected component containing $v_1$. If $v_3 \notin K$, swap
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colours $1$ and $3$ on $K$; now no neighbour of $v$ is coloured $1$, freeing it
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for $v$. If $v_3 \in K$, then a $1$--$3$ Kempe chain joins $v_1$ to $v_3$, and
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this chain together with $v$ encloses exactly one of $v_2, v_4$; hence the
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$2$--$4$ component containing $v_2$ cannot also reach $v_4$, and swapping colours
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$2$ and $4$ on it frees colour $2$ for $v$. Either way the colouring extends to
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$G$, a contradiction.
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Hence $\delta(G) \ge 5$.
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\end{proof}
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\section{The reduced dual}
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\label{sec:reduced-dual}
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Write $G'$ for the dual of $G$: since $G$ is a triangulation, $G'$ is a cubic
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plane graph in which each vertex of $G$ corresponds to a face of $G'$, each face
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of $G$ to a vertex of $G'$, and each edge to a dual edge. A vertex of $G$ of
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degree $k$ corresponds to a $k$-gonal face of $G'$.
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By Lemma~\ref{lem:mindeg}, $\delta(G) \ge 5$, and Euler's formula gives
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$\sum_{u \in V(G)}(6 - \deg u) = 12$, so $G$ has a vertex of degree exactly $5$
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(indeed at least twelve). Fix such a vertex $v$. Its dual face $F_v$ is a
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pentagon, bounded by the five dual vertices corresponding to the five faces of
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$G$ incident to $v$.
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\begin{definition}[Reduced dual]
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\label{def:reduced-dual}
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Let $v$ be a degree-$5$ vertex of $G$ with pentagonal dual face $F_v$, and fix an
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index $i \in \{0,1,2,3,4\}$. The \emph{reduced dual} $\widehat{G}'_{v,i}$ is the
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plane graph obtained from $G'$ as follows.
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\begin{enumerate}
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\item Delete the five dual vertices on the boundary of $F_v$, together with all
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edges incident to them. Each deleted vertex is cubic, with two edges on
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$\partial F_v$ and one edge leaving $F_v$; deleting the five boundary
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vertices therefore removes the five external edges as well, dropping their
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five outer endpoints from degree $3$ to degree $2$. These five degree-$2$
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vertices lie on the boundary of a single face $F$ of the resulting graph.
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\item List the five degree-$2$ vertices in clockwise order around $F$ as
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$A = (A_0, A_1, A_2, A_3, A_4)$.
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\item Add a new vertex $v_n$ and join it to $A_i$, $A_{i+1}$, and $A_{i+2}$
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(indices mod $5$) by three new edges.
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\item Add a new edge between $A_{i+3}$ and $A_{i+4}$ (indices mod $5$).
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\end{enumerate}
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\end{definition}
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\begin{remark}
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Steps (3) and (4) restore cubicity: $A_i, A_{i+1}, A_{i+2}$ each gain one edge to
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$v_n$ and $A_{i+3}, A_{i+4}$ each gain the new edge, so all five return to degree
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$3$, and $v_n$ has degree $3$. Since $A_i,\dots,A_{i+2}$ and $A_{i+3}, A_{i+4}$
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are each consecutive along $\partial F$, the new vertex and edge can be drawn
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inside $F$ without crossings, so $\widehat{G}'_{v,i}$ is again a cubic plane
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graph. The construction depends on the choice of $i$ up to the rotational
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symmetry of $A$.
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\end{remark}
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\begin{figure}[h]
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\centering
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\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step1.png}\hfill
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\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step2.png}\\[0.5em]
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\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step3.png}\hfill
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\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step4.png}
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\caption{The four steps of Definition~\ref{def:reduced-dual}, illustrated on
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$G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner
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pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$,
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leaving five degree-$2$ vertices on a new face $F$. Top right: order them
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clockwise as $A_0,\dots,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$.
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Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph
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$\widehat{G}'_{v,0}$.}
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\label{fig:reduced-dual-steps}
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\end{figure}
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\begin{definition}[Edges of the reduced dual]
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\label{def:edge-names}
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The four edges added in steps (3) and (4) of Definition~\ref{def:reduced-dual}
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are named as follows. The chord $A_{i+3}A_{i+4}$ is the \emph{merged edge};
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the edge $A_{i+1}v_n$ is the \emph{spike edge}; the edge $A_iv_n$ is the
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\emph{side-$0$ edge}; and the edge $A_{i+2}v_n$ is the \emph{side-$1$ edge}.
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In the $i = 0$ case of Figure~\ref{fig:reduced-dual-steps} these are
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$\{A_3, A_4\}$, $\{A_1, v_n\}$, $\{A_0, v_n\}$, and $\{A_2, v_n\}$
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respectively.
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\end{definition}
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We will use the following structural fact about proper $3$-edge-colourings near
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a pentagonal face of a cubic plane graph; it is stated for a generic such graph
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$H$, not specifically for the reduced dual.
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\begin{lemma}[Pentagonal externals]
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\label{lem:pentagonal-externals}
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Let $H$ be a cubic plane graph and $F$ a pentagonal face of $H$, with
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$\partial F$ traversed clockwise as $u_0, u_1, u_2, u_3, u_4$. For each $i$
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let $f_i$ be the unique edge of $H$ incident to $u_i$ that does not lie on
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$\partial F$. An assignment $\varphi$ of colours from $\{1, 2, 3\}$ to the ten
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edges incident to $\{u_0, \dots, u_4\}$ is proper at every $u_i$ if and only if
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there is some index $j$ such that
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\[
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\varphi(f_j) = \varphi(f_{j+1}) = \varphi(f_{j+2})
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\quad\text{and}\quad
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\{\varphi(f_{j+3}), \varphi(f_{j+4})\}
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= \{1, 2, 3\} \setminus \{\varphi(f_j)\},
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\]
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indices mod $5$.
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\end{lemma}
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\begin{proof}
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Write $e_i = u_i u_{i+1}$ for the boundary edges of $\partial F$ (indices mod
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$5$). A colouring $\varphi$ is proper at every $u_i$ if and only if at each
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$u_i$ the three incident edges $e_{i-1}, e_i, f_i$ receive three distinct
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colours; whenever this holds, $\varphi(f_i)$ is forced to be the unique colour
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in $\{1, 2, 3\} \setminus \{\varphi(e_{i-1}), \varphi(e_i)\}$, and $\varphi$
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restricts to a proper $3$-edge-colouring of the cycle $\partial F$.
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\textbf{($\Rightarrow$)} The line graph of $\partial F$ is $C_5$, whose
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maximum independent set has size $2$, so no colour appears more than twice on
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$\partial F$; and since $\partial F$ is an odd cycle, all three colours appear.
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The colour multiset on $(\varphi(e_0), \dots, \varphi(e_4))$ is therefore
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$(2, 2, 1)$, with the singleton at a unique position. Cyclically shifting
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indices we may place this position at $0$; let $c$ be the singleton colour.
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The remaining four edges form the path $e_1 e_2 e_3 e_4$, which by propriety
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alternates between the other two colours, so for some labelling
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$\{a, b, c\} = \{1, 2, 3\}$,
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\[
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(\varphi(e_0), \varphi(e_1), \varphi(e_2), \varphi(e_3), \varphi(e_4))
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= (c, a, b, a, b).
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\]
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Reading off the forced values of $\varphi(f_i)$,
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\[
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\varphi(f_0) = a, \quad
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\varphi(f_1) = b, \quad
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\varphi(f_2) = \varphi(f_3) = \varphi(f_4) = c,
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\]
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which is the lemma's pattern at $j = 2$ (the cyclic shift maps this back to
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the corresponding $j$ in the original indexing). This case is the unique
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proper $3$-edge-colouring of $\partial F$ up to cyclic shift and permutation of
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$\{1, 2, 3\}$ (since $5 \cdot 3! = 30 = P(C_5, 3)$, the chromatic polynomial of
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$C_5$ at $3$), so it exhausts every proper $\varphi$.
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\textbf{($\Leftarrow$)} The lemma's hypothesis is invariant under cyclic
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shifts of indices and under permutations of $\{1, 2, 3\}$, so we may assume
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$j = 2$, $\varphi(f_2) = \varphi(f_3) = \varphi(f_4) = c$, $\varphi(f_0) = a$,
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and $\varphi(f_1) = b$, with $\{a, b, c\} = \{1, 2, 3\}$. Propriety at $u_i$
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and $u_{i+1}$ requires $\varphi(e_i) \notin \{\varphi(f_i), \varphi(f_{i+1})\}$,
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which gives
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\[
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\varphi(e_0) = c, \quad
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\varphi(e_1) = a, \quad
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\varphi(e_2) \in \{a, b\}, \quad
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\varphi(e_3) \in \{a, b\}, \quad
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\varphi(e_4) = b.
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\]
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The remaining propriety condition $\varphi(e_{i-1}) \neq \varphi(e_i)$ holds
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automatically at $u_0, u_1, u_4$, forces $\varphi(e_2) = b$ at $u_2$, and then
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forces $\varphi(e_3) = a$ at $u_3$. The resulting triples
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$(\varphi(e_{i-1}), \varphi(e_i), \varphi(f_i))$ at $u_0, u_1, u_2, u_3, u_4$
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are
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\[
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(b, c, a), \quad (c, a, b), \quad (a, b, c), \quad (b, a, c), \quad (a, b, c),
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\]
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each a permutation of $\{1, 2, 3\}$, so $\varphi$ is proper at every $u_i$.
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\end{proof}
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\begin{remark}
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The two-element condition $\{\varphi(f_{j+3}), \varphi(f_{j+4})\}
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= \{1,2,3\}\setminus\{\varphi(f_j)\}$ cannot be dropped: a 3-colouring
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satisfying $\varphi(f_j) = \varphi(f_{j+1}) = \varphi(f_{j+2})$ alone need not
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extend, e.g.\ $(1,1,1,1,2)$.
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\end{remark}
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Since $\widehat{G}'_{v,i}$ is the dual of a triangulation on fewer vertices than
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$G$, it is $3$-edge-colourable by the minimality of $G$. The following lemma
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constrains every such colouring.
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\begin{lemma}
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\label{lem:chord-apex}
|
|
Let $G$ be a minimal counterexample, and let $\widehat{G}'_{v,i}$ be a reduced
|
|
dual of its dual $G'$. Then in every proper $3$-edge-colouring of
|
|
$\widehat{G}'_{v,i}$, the merged edge and the spike edge receive the same
|
|
colour.
|
|
\end{lemma}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{fig_chord_apex_step1.png}\\[0.4em]
|
|
\includegraphics[width=0.48\textwidth]{fig_chord_apex_step2.png}\hfill
|
|
\includegraphics[width=0.48\textwidth]{fig_chord_apex_step3.png}
|
|
\caption{The proof of Lemma~\ref{lem:chord-apex}, illustrated for $i = 0$ on
|
|
$G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at
|
|
$v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has
|
|
externals satisfying the hypothesis of
|
|
Lemma~\ref{lem:pentagonal-externals}, which colours $\partial F_v$ to extend
|
|
$\psi$ to a proper $3$-edge-colouring of $G'$.}
|
|
\label{fig:chord-apex-proof}
|
|
\end{figure}
|
|
|
|
\begin{proof}
|
|
After cyclically relabelling, assume $i = 0$. Suppose for contradiction that
|
|
$\varphi$ is a proper $3$-edge-colouring of $\widehat{G}'_{v,0}$ in which the
|
|
merged edge $\{A_3, A_4\}$ and the spike edge $\{A_1, v_n\}$ receive different
|
|
colours (Figure~\ref{fig:chord-apex-proof}, top), and write
|
|
\[
|
|
X = \varphi(\{A_0, v_n\}), \quad
|
|
Y = \varphi(\{A_1, v_n\}), \quad
|
|
Z = \varphi(\{A_2, v_n\}), \quad
|
|
W = \varphi(\{A_3, A_4\}).
|
|
\]
|
|
Propriety of $\varphi$ at $v_n$ forces $\{X, Y, Z\} = \{1, 2, 3\}$, and the
|
|
assumption $W \neq Y$ leaves $W \in \{X, Z\}$.
|
|
|
|
We lift $\varphi$ to a colouring $\psi$ of $E(G')$ as follows. Let
|
|
$B_0, \dots, B_4$ be the boundary vertices of $\partial F_v$ in $G'$, indexed
|
|
so that $f_k = B_k A_k$. On every edge of $G'$ that survived the reduction,
|
|
set $\psi = \varphi$. At each $A_k$ the two surviving edges retain their
|
|
$\varphi$-colours, so the remaining edge at $A_k$ --- in $G'$ this is the
|
|
external $f_k$; in $\widehat{G}'_{v,0}$ this is a $v_n$-edge ($k \in
|
|
\{0, 1, 2\}$) or the chord ($k \in \{3, 4\}$) --- is forced to take the third
|
|
colour at $A_k$. Since the two-surviving-edge colours at $A_k$ agree in $G'$
|
|
and $\widehat{G}'_{v,0}$, the third colour does too, giving
|
|
\[
|
|
\psi(f_0) = X, \quad \psi(f_1) = Y, \quad \psi(f_2) = Z, \quad
|
|
\psi(f_3) = \psi(f_4) = W
|
|
\]
|
|
(the last two equalities holding because the chord is a single edge contributing
|
|
its colour at each of $A_3$ and $A_4$).
|
|
|
|
It remains to assign colours to the five boundary edges $B_k B_{k+1}$ of
|
|
$\partial F_v$. Apply Lemma~\ref{lem:pentagonal-externals} to $G'$ at the face
|
|
$F_v$ with the $B_k$'s as its boundary vertices and the same indexing. The
|
|
external vector $(\psi(f_0), \dots, \psi(f_4)) = (X, Y, Z, W, W)$ falls into
|
|
one of two cases (Figure~\ref{fig:chord-apex-proof}, bottom):
|
|
\begin{itemize}
|
|
\item if $W = Z$, it is $(X, Y, Z, Z, Z)$: three consecutive $Z$'s at
|
|
positions $2, 3, 4$, with $\{X, Y\} = \{1, 2, 3\} \setminus \{Z\}$;
|
|
\item if $W = X$, it is $(X, Y, Z, X, X)$: three consecutive $X$'s at
|
|
positions $3, 4, 0$, with $\{Y, Z\} = \{1, 2, 3\} \setminus \{X\}$.
|
|
\end{itemize}
|
|
Each case satisfies the hypothesis of Lemma~\ref{lem:pentagonal-externals};
|
|
its $(\Leftarrow)$ direction therefore assigns colours to the boundary edges
|
|
$B_k B_{k+1}$ that make $\psi$ proper at every $B_k$.
|
|
|
|
The resulting $\psi$ is a proper $3$-edge-colouring of $G'$: proper at every
|
|
$B_k$ by the lemma, at every $A_k$ by the construction, and at every other
|
|
vertex because such a vertex has the same neighbourhood in $G'$ as in
|
|
$\widehat{G}'_{v,0}$ with the same incident-edge colours. By Tait's theorem,
|
|
$G'$ is $3$-edge-colourable iff $G$ is $4$-vertex-colourable, contradicting
|
|
that $G$ is a counterexample. The assumption $W \neq Y$ is therefore false.
|
|
\end{proof}
|
|
|
|
For a pair of colours $\{a, b\} \subseteq \{1, 2, 3\}$, the subgraph of
|
|
$\widehat{G}'_{v,i}$ on the edges coloured $a$ or $b$ is $2$-regular (since at
|
|
each vertex exactly one of the three incident edges is excluded), and hence a
|
|
disjoint union of cycles. We call each such cycle a \emph{$\{a, b\}$-Kempe
|
|
cycle}, and reserve the notation for the specific cycle containing a given
|
|
edge when the context makes it clear. Swapping the two colours on a single
|
|
Kempe cycle yields another proper $3$-edge-colouring of the same graph.
|
|
|
|
\begin{lemma}[Kempe cycles through the spike]
|
|
\label{lem:kempe-spike}
|
|
Let $G$ be a minimal counterexample, fix a reduced dual $\widehat{G}'_{v,i}$ of
|
|
$G'$, and let $\varphi$ be a proper $3$-edge-colouring of $\widehat{G}'_{v,i}$.
|
|
Write $c$ for the common colour assigned by $\varphi$ to the spike and the
|
|
merged edge (Lemma~\ref{lem:chord-apex}), and $c_0, c_1$ for the colours of
|
|
the side-$0$ and side-$1$ edges respectively, so $\{c, c_0, c_1\} = \{1, 2,
|
|
3\}$. Then
|
|
\begin{enumerate}
|
|
\item the $\{c, c_0\}$-Kempe cycle through the spike edge contains both the
|
|
side-$0$ edge and the merged edge;
|
|
\item the $\{c, c_1\}$-Kempe cycle through the spike edge contains both the
|
|
side-$1$ edge and the merged edge.
|
|
\end{enumerate}
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
We prove (1); (2) is the same argument with $c_1$ and the side-$1$ edge in
|
|
place of $c_0$ and the side-$0$ edge.
|
|
|
|
The spike edge $\{A_{i+1}, v_n\}$ and the side-$0$ edge $\{A_i, v_n\}$ share
|
|
the vertex $v_n$ and receive the two colours $c, c_0$, so they both lie on the
|
|
$\{c, c_0\}$-Kempe cycle through $v_n$. Suppose for contradiction that the
|
|
merged edge lies on a different $\{c, c_0\}$-Kempe cycle $K$ (it lies on
|
|
\emph{some} such cycle, since it has colour $c$). Let $\varphi'$ be obtained
|
|
from $\varphi$ by swapping the colours $c$ and $c_0$ along $K$ alone: this is
|
|
a Kempe swap, so $\varphi'$ is again a proper $3$-edge-colouring of
|
|
$\widehat{G}'_{v,i}$. Under $\varphi'$ the spike edge --- which is not on $K$
|
|
--- still has colour $c$, but the merged edge --- which is on $K$ --- now has
|
|
colour $c_0$. Hence in $\varphi'$ the spike and the merged edge receive
|
|
distinct colours, contradicting Lemma~\ref{lem:chord-apex} applied to
|
|
$\varphi'$.
|
|
\end{proof}
|
|
|
|
\section{Edge suppression}
|
|
\label{sec:edge-suppression}
|
|
|
|
We recall the classical operation of \emph{edge suppression} on cubic
|
|
plane graphs: delete the edge and smooth the two resulting degree-$2$
|
|
endpoints. Under planar duality this coincides with simple-graph
|
|
contraction on the dual side. It will be the central tool in
|
|
Section~\ref{sec:toward-4ct} below, where we formulate a sufficient
|
|
condition for the Four Colour Theorem.
|
|
|
|
\begin{definition}[Edge suppression]
|
|
\label{def:edge-suppression}
|
|
Let $H$ be a cubic plane graph and $e = uv$ an edge of $H$ with $u \neq v$
|
|
and no edge of $H$ parallel to $e$. The \emph{edge suppression} of $H$ at
|
|
$e$ is the graph $H'$ obtained in two steps:
|
|
\begin{enumerate}
|
|
\item \emph{Delete} the edge $e$; the endpoints $u$ and $v$ each drop to
|
|
degree $2$.
|
|
\item \emph{Smooth} each of $u$ and $v$: at $u$, replace $u$ and its two
|
|
remaining incident edges $ua, ub$ by a single new edge $ab$; do
|
|
the same at $v$. Both vertices $u$ and $v$ are removed, and two
|
|
new edges are added in their place.
|
|
\end{enumerate}
|
|
Provided the smoothings do not introduce a loop or parallel edge, $H'$ is
|
|
again a cubic plane graph, with $|V(H')| = |V(H)| - 2$ and
|
|
$|E(H')| = |E(H)| - 3$.
|
|
|
|
Equivalently, $H'$ is the planar dual of $\mathrm{dual}(H) / e^{*}$,
|
|
where $e^{*}$ is the edge of $\mathrm{dual}(H)$ crossing $e$ and the
|
|
contraction on the right-hand side is simple-graph contraction (loops
|
|
removed, parallel edges absorbed). Under planar duality, contracting
|
|
$e^{*}$ in $\mathrm{dual}(H)$ merges the two triangular faces of
|
|
$\mathrm{dual}(H)$ incident to $e^{*}$, and the parallel-edge cleanup
|
|
corresponds exactly to the smoothing step on the primal side.
|
|
\end{definition}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=0.95\textwidth]{fig_cubic_edge_contraction.png}
|
|
\caption{Edge suppression
|
|
(Definition~\ref{def:edge-suppression}). Left: a fragment of a cubic
|
|
plane graph with the suppressed edge $e = uv$ highlighted in red. Middle:
|
|
deleting $e$ leaves $u$ and $v$ of degree~$2$. Right: smoothing $u$ and
|
|
$v$ replaces each pair of incident edges by a single new edge, removing
|
|
$u, v$ and giving a cubic plane graph again.}
|
|
\label{fig:edge-suppression}
|
|
\end{figure}
|
|
|
|
\begin{theorem}[Edge suppression across a 4-face preserves 3-edge-colourability]
|
|
\label{thm:edge-suppression-4face}
|
|
Let $H$ be a cubic plane graph with a proper $3$-edge-colouring $\varphi$,
|
|
let $f$ be a face of $H$ with $|\partial f| = 4$, and let $e_0, e_1$ be
|
|
the two edges of $\partial f$ sharing no endpoint (the opposite pair on
|
|
the $4$-cycle $\partial f$). If $\varphi(e_0) \neq \varphi(e_1)$ and the
|
|
edge suppression of $H$ at $e_0$
|
|
(Definition~\ref{def:edge-suppression}) is well-defined (no loops or
|
|
parallel edges are created), then the suppressed graph admits a proper
|
|
$3$-edge-colouring.
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
Write $\partial f$ as the $4$-cycle $v_0 v_1 v_2 v_3$ with $e_0 = v_0 v_1$
|
|
and $e_1 = v_2 v_3$ (so $e_0, e_1$ are opposite); the remaining two
|
|
boundary edges of $f$ are $e_2 := v_1 v_2$ and $e_3 := v_3 v_0$. Since $H$
|
|
is cubic, each $v_i$ has exactly one edge not on $\partial f$: write $w_i$
|
|
for that edge and $u_i$ for its other endpoint, so $w_i = v_i u_i$ with
|
|
$u_i \notin \{v_0, v_1, v_2, v_3\}$, for each $i \in \{0, 1, 2, 3\}$. Put
|
|
$a := \varphi(e_0)$, $b := \varphi(e_1)$, and let $c$ be the third colour.
|
|
|
|
\emph{Forced colours on the face.} Propriety at $v_1$ and $v_2$ forces
|
|
$\varphi(e_2) \notin \{a, b\}$, so $\varphi(e_2) = c$; then
|
|
$\varphi(w_1) = b$ and $\varphi(w_2) = a$. Symmetrically $\varphi(e_3) = c$,
|
|
$\varphi(w_0) = b$, and $\varphi(w_3) = a$. In particular
|
|
$\varphi(w_0) = \varphi(w_1) = b$.
|
|
|
|
\emph{Construction of $\varphi'$.} Let $H'$ denote the edge suppression
|
|
of $H$ at $e_0$; its new edges are $e_3' := v_3 u_0$
|
|
(replacing $e_3$ and $w_0$ via the smoothing at $v_0$) and
|
|
$e_2' := v_2 u_1$ (replacing $e_2$ and $w_1$ via the smoothing at $v_1$).
|
|
Define $\varphi' \colon E(H') \to \{1, 2, 3\}$ by
|
|
\[
|
|
\varphi'(e) :=
|
|
\begin{cases}
|
|
c & \text{if } e = e_1, \\
|
|
b & \text{if } e \in \{e_2', e_3'\}, \\
|
|
\varphi(e) & \text{otherwise.}
|
|
\end{cases}
|
|
\]
|
|
That is: give each smoothed-in edge the colour $b$ (the colour of the two
|
|
$w_i$ it absorbs), recolour $e_1$ to $c$, and leave every other edge of
|
|
$H'$ with its $\varphi$-colour.
|
|
|
|
\emph{Propriety.} Every vertex of $H'$ other than $v_2, v_3, u_0, u_1$ has
|
|
the same incident edges and the same $\varphi'$-colours as it did under
|
|
$\varphi$, so propriety is inherited there. At the four affected vertices,
|
|
\[
|
|
\begin{array}{r|lll}
|
|
\text{vertex} & \text{edges in } H' & \text{colours under }\varphi' \\
|
|
\hline
|
|
v_2 & e_1,\; w_2,\; e_2' & c,\; a,\; b \\
|
|
v_3 & e_1,\; w_3,\; e_3' & c,\; a,\; b \\
|
|
u_0 & e_3',\; \alpha_0,\; \beta_0 & b,\; a,\; c \\
|
|
u_1 & e_2',\; \alpha_1,\; \beta_1 & b,\; a,\; c
|
|
\end{array}
|
|
\]
|
|
where $\alpha_i, \beta_i$ are the two edges of $H$ at $u_i$ other than
|
|
$w_i$, whose $\varphi$-colours are forced to $\{a, c\}$ by propriety at
|
|
$u_i$ (since $\varphi(w_i) = b$). Each row lists three distinct colours, so
|
|
$\varphi'$ is proper.
|
|
\end{proof}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=0.98\textwidth]{fig_thm_cubic_contraction_4face.png}
|
|
\caption{The recolouring used in the proof of
|
|
Theorem~\ref{thm:edge-suppression-4face}. Left: the $4$-face $f$ of $H$
|
|
under $\varphi$, with the forced colours $\varphi(e_0) = a$,
|
|
$\varphi(e_1) = b$, $\varphi(e_2) = \varphi(e_3) = c$,
|
|
$\varphi(w_0) = \varphi(w_1) = b$, and $\varphi(w_2) = \varphi(w_3) = a$.
|
|
Right: the suppressed graph $H'$ under $\varphi'$. The smoothed-in edges
|
|
$e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is
|
|
recoloured from $b$ to $c$; every edge outside the face neighbourhood
|
|
keeps its $\varphi$-colour (dotted in red: the five edges of $H$ removed
|
|
by the suppression).}
|
|
\label{fig:thm-edge-suppression-4face}
|
|
\end{figure}
|
|
|
|
\section{The face-monochromatic-pair conjecture and the Four Colour Theorem}
|
|
\label{sec:toward-4ct}
|
|
|
|
The following conjecture identifies a structural property of every proper
|
|
$3$-edge-colouring of a reduced dual of a minimal counterexample. If true,
|
|
it implies the Four Colour Theorem via
|
|
Theorem~\ref{thm:edge-suppression-4face}.
|
|
|
|
\begin{conjecture}[Face-monochromatic-pair conjecture]
|
|
\label{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
Let $G$ be a minimal counterexample to the Four Colour Theorem, and let
|
|
$\widehat{G}'_{v,i}$ be a reduced dual of $G' = \mathrm{dual}(G)$. Then
|
|
for every proper $3$-edge-colouring $\varphi$ of $\widehat{G}'_{v,i}$
|
|
there exist a face $F$ of $\widehat{G}'_{v,i}$ and two distinct edges
|
|
$e_1, e_2 \in \partial F$, with neither $e_1$ nor $e_2$ equal to the
|
|
merged edge, such that:
|
|
\begin{enumerate}
|
|
\item $\varphi(e_1) = \varphi(e_2)$. Write
|
|
$a := \varphi(e_1) = \varphi(e_2)$.
|
|
\item $e_1$, $e_2$, and the merged edge all lie on a common
|
|
$\{a, b\}$-Kempe cycle of $\varphi$, for some colour $b \neq a$.
|
|
\item Exactly one edge of $\partial F$ lies between $e_1$ and $e_2$
|
|
along one of the two arcs of $\partial F$. Equivalently,
|
|
subdividing $e_1, e_2$ by new vertices $X_1, X_2$ and joining
|
|
them by a new edge $X_1 X_2$ inside $F$ creates a new face $f_n$
|
|
bounded by exactly $4$ edges (the new edge $X_1 X_2$, the two
|
|
subdivision halves adjacent to it, and the single
|
|
$\partial F$-edge between $e_1$ and $e_2$).
|
|
\end{enumerate}
|
|
\end{conjecture}
|
|
|
|
\begin{remark}
|
|
\label{rem:conj-3-6-empirical}
|
|
\sloppy
|
|
The conjecture cannot be tested on actual minimal counterexamples (none
|
|
exist by the Four Colour Theorem), but its conclusion is checkable on the
|
|
structural surrogates: proper $3$-edge-colourings of reduced duals that
|
|
satisfy both the chord-apex condition (Lemma~\ref{lem:chord-apex}) and
|
|
the Kempe-cycle conditions (Lemma~\ref{lem:kempe-spike}), since a minimal
|
|
counterexample's reduced dual is forced to admit such colourings. For
|
|
every min-degree-$5$ triangulation $G$ with $|V(G)| \leq 21$, every
|
|
pentagonal face $F$ of $G'$, and every reduction index
|
|
$i \in \{0,\dots,4\}$, we enumerated all such colourings and verified the
|
|
three clauses of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} (see
|
|
\texttt{experiments/check\_conj\_face\_kempe\_scaled.py}); $n = 22$ ran
|
|
past a $1800$\,s budget after $641{,}700$ colourings (all pass) without
|
|
finishing the full set of $651$ triangulations.
|
|
\begin{center}
|
|
\small
|
|
\renewcommand{\arraystretch}{1.15}
|
|
\begin{tabular}{r|r|r|r|l}
|
|
$n$ & \#tri & \#col.\ tested & \#sat. & status \\
|
|
\hline
|
|
$12$ & $1$ & $0$ & --- & vacuous (icosahedron) \\
|
|
$13$ & $0$ & --- & --- & no min-deg-$5$ tri \\
|
|
$14$ & $1$ & $216$ & $216$ & all pass \\
|
|
$15$ & $1$ & $0$ & --- & vacuous \\
|
|
$16$ & $3$ & $864$ & $864$ & all pass \\
|
|
$17$ & $4$ & $4{,}650$ & $4{,}650$ & all pass \\
|
|
$18$ & $12$ & $8{,}070$ & $8{,}070$ & all pass \\
|
|
$19$ & $23$ & $21{,}138$ & $21{,}138$ & all pass \\
|
|
$20$ & $73$ & $107{,}874$ & $107{,}874$ & all pass \\
|
|
$21$ & $192$ & $392{,}370$ & $392{,}370$ & all pass \\
|
|
$22$ (part.) & $651$ & $641{,}700$ & $641{,}700$ & timeout \\
|
|
\hline
|
|
total ($n \le 21$) & $311$ & $535{,}182$ & $535{,}182$ & \\
|
|
\end{tabular}
|
|
\end{center}
|
|
\noindent The vacuous rows ($n = 12, 15$) are those where the relevant
|
|
reduced duals admit no chord-apex+Kempe colourings, so the conjecture
|
|
has no content there.
|
|
\end{remark}
|
|
|
|
The following strengthening adds a fourth clause that arranges the new
|
|
$4$-face $f_n$ to satisfy the hypotheses of
|
|
Theorem~\ref{thm:edge-suppression-4face}.
|
|
|
|
\begin{conjecture}[Strengthening of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}]
|
|
\label{conj:face-monochromatic-pair-strengthened}
|
|
Let $G$, $\widehat{G}'_{v,i}$, $\varphi$ be as in
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}. Then
|
|
there exist $F$, $e_1$, $e_2$ satisfying clauses~(1)--(3) of that
|
|
conjecture, together with the following additional clause.
|
|
\begin{enumerate}
|
|
\setcounter{enumi}{3}
|
|
\item Let $\widehat{G}'^{+}$ be the modified graph obtained from
|
|
$\widehat{G}'_{v,i}$ by the modification of clause~(3), and let
|
|
$\varphi'$ be the proper $3$-edge-colouring of $\widehat{G}'^{+}$
|
|
obtained from $\varphi$ by swapping the two colours along the
|
|
(subdivided) $\{a, b\}$-Kempe cycle of clause~(2) and assigning
|
|
the new edge $X_1 X_2$ the remaining (third) colour $c$.
|
|
(Equivalently: $\varphi'$ agrees with $\varphi$ on every edge of
|
|
$\widehat{G}'_{v,i}$ outside that Kempe cycle, and at $X_1, X_2$
|
|
the two subdivision halves take the colours $\{a, b\}$ in the
|
|
order forced by propriety.) Then either
|
|
\begin{enumerate}
|
|
\item[(i)] $\partial f_n$ uses all three colours under
|
|
$\varphi'$, or
|
|
\item[(ii)] the $\{b, c\}$-Kempe cycle of $\varphi'$ through
|
|
$X_1 X_2$ is incident to exactly one edge of
|
|
$\partial f_n$ (namely $X_1 X_2$ itself).
|
|
\end{enumerate}
|
|
\end{enumerate}
|
|
\end{conjecture}
|
|
|
|
\begin{remark}
|
|
\label{rem:conj-3-8-empirical}
|
|
\sloppy
|
|
The strengthened conjecture was tested on the same chord-apex+Kempe
|
|
colourings as Remark~\ref{rem:conj-3-6-empirical}; for each colouring we
|
|
sought any
|
|
Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}-witness
|
|
$(F, e_1, e_2)$ whose accompanying $f_n$ satisfies clause~(4) (see
|
|
\texttt{experiments/check\_conj\_3\_8\_scaled.py}):
|
|
\begin{center}
|
|
\small
|
|
\renewcommand{\arraystretch}{1.15}
|
|
\begin{tabular}{r|r|r|r|l}
|
|
$n$ & \#tri & \#col.\ tested & \#sat. & status \\
|
|
\hline
|
|
$14$ & $1$ & $216$ & $216$ & all pass \\
|
|
$16$ & $3$ & $864$ & $864$ & all pass \\
|
|
$17$ & $4$ & $4{,}650$ & $4{,}650$ & all pass \\
|
|
$18$ & $12$ & $8{,}070$ & $8{,}070$ & all pass \\
|
|
$19$ & $23$ & $21{,}138$ & $21{,}138$ & all pass \\
|
|
$20$ & $73$ & $107{,}874$ & $107{,}874$ & all pass \\
|
|
\hline
|
|
total ($n \le 20$) & $116$ & $142{,}812$ & $142{,}812$ & \\
|
|
\end{tabular}
|
|
\end{center}
|
|
\noindent A subtlety: only about half of the
|
|
Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}-witnesses
|
|
individually satisfy clause~(4) on each colouring, but in every case
|
|
\emph{some} witness does. Clause~(4) is therefore an existential statement
|
|
at the witness level, not a property of every witness.
|
|
|
|
\smallskip\noindent\emph{Targeted check on the Holton--McKay duals.}
|
|
The six $21$-vertex triangulations whose duals are the
|
|
non-Hamiltonian $38$-vertex cubic plane graphs of Holton and McKay
|
|
(the smallest examples falsifying Tait's conjecture) are a particularly
|
|
interesting subfamily at $n = 21$. Running the strengthened test directly
|
|
on these six (see
|
|
\texttt{experiments/check\_conj\_on\_holton\_mckay.py}) gives:
|
|
|
|
\begin{center}
|
|
\small
|
|
\renewcommand{\arraystretch}{1.15}
|
|
\begin{tabular}{r|r|r|r|l}
|
|
HM\# & \#pentagonal faces & \#col.\ tested & \#sat.\ (1)--(4) & status \\
|
|
\hline
|
|
$0$ & $10$ & $2{,}880$ & $2{,}880$ & all pass \\
|
|
$1$ & $11$ & $2{,}880$ & $2{,}880$ & all pass \\
|
|
$2$ & $10$ & $2{,}880$ & $2{,}880$ & all pass \\
|
|
$3$ & $10$ & $2{,}880$ & $2{,}880$ & all pass \\
|
|
$4$ & $11$ & $2{,}880$ & $2{,}880$ & all pass \\
|
|
$5$ & $11$ & $2{,}880$ & $2{,}880$ & all pass \\
|
|
\hline
|
|
total & --- & $17{,}280$ & $17{,}280$ & \\
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{remark}
|
|
|
|
\begin{remark}[The implication to the Four Colour Theorem]
|
|
\label{rem:implication-4ct}
|
|
Clause~(4)(i) of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-strengthened} says that
|
|
$\partial f_n$ uses all three colours under $\varphi'$. Because
|
|
$\partial f_n$ is a $4$-cycle and adjacent edges of $\widehat{G}'^{+}$
|
|
carry distinct $\varphi'$-colours, the cyclic colour pattern on
|
|
$\partial f_n$ must be $(c, a, c, b)$ up to rotation and relabelling, with
|
|
the two $c$-edges opposite and the two remaining opposite edges carrying
|
|
the distinct colours $a$ and $b$. Those two opposite edges therefore
|
|
satisfy the hypothesis of Theorem~\ref{thm:edge-suppression-4face}: they
|
|
lie on the $4$-face $f_n$, share no endpoint, and have different
|
|
$\varphi'$-colours. Theorem~\ref{thm:edge-suppression-4face} then produces
|
|
a proper $3$-edge-colouring of the edge suppression of
|
|
$\widehat{G}'^{+}$ at the $a$-coloured one.
|
|
|
|
Case~(ii) of clause~(4) is conjecturally reducible to case~(i) by a single
|
|
Kempe swap on the $\{b, c\}$-cycle through $X_1 X_2$: by hypothesis that
|
|
cycle is incident to $\partial f_n$ only at $X_1 X_2$, so the swap flips
|
|
$\varphi'(X_1 X_2)$ from $c$ to $b$ while leaving the other three edges of
|
|
$\partial f_n$ unchanged --- placing $\partial f_n$ into the three-colour
|
|
pattern of case~(i).
|
|
|
|
\smallskip\noindent\emph{Consequence.}
|
|
Theorem~\ref{thm:edge-suppression-4face} now produces a proper
|
|
$3$-edge-colouring of the edge suppression of $\widehat{G}'^{+}$ at the
|
|
chosen edge of $f_n$. Combined with the
|
|
chord-apex and Kempe-cycle structure of $\widehat{G}'_{v,i}$
|
|
(Lemmas~\ref{lem:chord-apex} and~\ref{lem:kempe-spike}), this yields a
|
|
proper $3$-edge-colouring of $G'$, and by Tait's correspondence a proper
|
|
$4$-vertex-colouring of $G$ --- contradicting the assumption that $G$ is a
|
|
minimal counterexample. Hence
|
|
Conjecture~\ref{conj:face-monochromatic-pair-strengthened} implies the
|
|
Four Colour Theorem.
|
|
\end{remark}
|
|
|
|
\begin{thebibliography}{9}
|
|
\bibitem{AH77a}
|
|
K.~Appel and W.~Haken,
|
|
\emph{Every planar map is four colorable. Part~I: Discharging},
|
|
Illinois J.~Math. \textbf{21} (1977), 429--490.
|
|
|
|
\bibitem{AHK77}
|
|
K.~Appel, W.~Haken, and J.~Koch,
|
|
\emph{Every planar map is four colorable. Part~II: Reducibility},
|
|
Illinois J.~Math. \textbf{21} (1977), 491--567.
|
|
|
|
\bibitem{RSST97}
|
|
N.~Robertson, D.~P.~Sanders, P.~Seymour, and R.~Thomas,
|
|
\emph{The four-colour theorem},
|
|
J.~Combin. Theory Ser.~B \textbf{70} (1997), 2--44.
|
|
|
|
\bibitem{Gonthier08}
|
|
G.~Gonthier,
|
|
\emph{Formal proof---the four-color theorem},
|
|
Notices Amer. Math. Soc. \textbf{55} (2008), no.~11, 1382--1393.
|
|
\end{thebibliography}
|
|
|
|
\end{document}
|