didericis
cfc53dcbe2
Follow-up to Lemma 5.2. States that if Conjecture 5.1 has no
clauses-(1)-(3) witness for (G, G'^_{v,i}, phi), then h_phi is
constant on both Kempe cycles through merged, and the two constants
agree (since merged is on both cycles, so its endpoints force the
constants to match).
Proof is the V1-direction of the case analysis: differing h_phi on
either K_b or K_c reproduces a clause-(1)-(3) witness by the same
F_R/F_L geometry as Lemma 5.2's proof but with the hypothesis
"h_phi(v_0) != h_phi(v_1)", under which the matching-colour edges
land on the SAME face of e. Case B's merged-incidence corner is
handled by choosing a differing-Heawood pair away from merged's
endpoints.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
995 lines
46 KiB
TeX
995 lines
46 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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The following labelling of vertices in a properly $3$-edge-coloured cubic
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plane graph will be useful in Section~\ref{sec:toward-4ct}.
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\begin{definition}[Heawood number of a vertex]
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\label{def:heawood-number}
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Let $H$ be a cubic plane graph with a fixed planar embedding, and let
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$\varphi \colon E(H) \to \{1, 2, 3\}$ be a proper $3$-edge-colouring. At
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each vertex $v \in V(H)$, the three incident edges receive three distinct
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colours; reading them in clockwise order around $v$ gives a cyclic
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permutation of $(1, 2, 3)$. The \emph{Heawood number} of $v$ is
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\[
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h_\varphi(v) :=
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\begin{cases}
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+1 & \text{if the clockwise cyclic colour order at $v$ is }(1, 2, 3), \\
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-1 & \text{if it is }(1, 3, 2).
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\end{cases}
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\]
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Equivalently, $h_\varphi(v) = +1$ when the clockwise colour order at $v$ is
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an even cyclic permutation of $(1, 2, 3)$ and $-1$ when it is an odd one.
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The labels are due to Heawood~\cite{Heawood1898}, who introduced them as
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part of his analysis of $3$-edge-colourings of cubic plane graphs.
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\end{definition}
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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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\[
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\varphi(e_0) = c, \quad
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\varphi(e_1) = a, \quad
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\varphi(e_2) \in \{a, b\}, \quad
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\varphi(e_3) \in \{a, b\}, \quad
|
|
\varphi(e_4) = b.
|
|
\]
|
|
The remaining propriety condition $\varphi(e_{i-1}) \neq \varphi(e_i)$ holds
|
|
automatically at $u_0, u_1, u_4$, forces $\varphi(e_2) = b$ at $u_2$, and then
|
|
forces $\varphi(e_3) = a$ at $u_3$. The resulting triples
|
|
$(\varphi(e_{i-1}), \varphi(e_i), \varphi(f_i))$ at $u_0, u_1, u_2, u_3, u_4$
|
|
are
|
|
\[
|
|
(b, c, a), \quad (c, a, b), \quad (a, b, c), \quad (b, a, c), \quad (a, b, c),
|
|
\]
|
|
each a permutation of $\{1, 2, 3\}$, so $\varphi$ is proper at every $u_i$.
|
|
\end{proof}
|
|
|
|
\begin{remark}
|
|
The two-element condition $\{\varphi(f_{j+3}), \varphi(f_{j+4})\}
|
|
= \{1,2,3\}\setminus\{\varphi(f_j)\}$ cannot be dropped: a 3-colouring
|
|
satisfying $\varphi(f_j) = \varphi(f_{j+1}) = \varphi(f_{j+2})$ alone need not
|
|
extend, e.g.\ $(1,1,1,1,2)$.
|
|
\end{remark}
|
|
|
|
Since $\widehat{G}'_{v,i}$ is the dual of a triangulation on fewer vertices than
|
|
$G$, it is $3$-edge-colourable by the minimality of $G$. The following lemma
|
|
constrains every such colouring.
|
|
|
|
\begin{lemma}
|
|
\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{lemma}[A Heawood-constant Kempe cycle does not admit the clause-(3) arc]
|
|
\label{lem:kempe-heawood-constant}
|
|
Let $G$ be a minimal counterexample to the Four Colour Theorem, fix a
|
|
reduced dual $\widehat{G}'_{v,i}$ of $G' = \mathrm{dual}(G)$, and let
|
|
$\varphi$ be a proper $3$-edge-colouring of $\widehat{G}'_{v,i}$. Set
|
|
$a := \varphi(\mathrm{merged})$ and let $K$ be the $\{a, b\}$-Kempe
|
|
cycle of $\varphi$ through the merged edge for some
|
|
$b \in \{1, 2, 3\} \setminus \{a\}$. If $h_\varphi$ is constant on
|
|
$V(K)$, then no edge $e \in E(K)$ admits a face $F$ of
|
|
$\widehat{G}'_{v,i}$ and two non-incident edges
|
|
$e_1, e_2 \in \partial F$ such that
|
|
$\varphi(e_1) = \varphi(e_2)$ and $e$ is the unique edge of
|
|
$\partial F$ between $e_1$ and $e_2$ along one of the two arcs of
|
|
$\partial F$ --- that is, no edge of $K$ admits the clause-(3) arc of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
with $e_1, e_2$ at its two endpoints.
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
Let $c$ be the third colour. Fix any edge $e \in E(K)$ joining
|
|
$v_0, v_1 \in V(K)$. By hypothesis $h_\varphi(v_0) = h_\varphi(v_1)$;
|
|
after possibly relabelling we may take
|
|
$h_\varphi(v_0) = h_\varphi(v_1) = +1$, so by
|
|
Definition~\ref{def:heawood-number} the clockwise cyclic colour order
|
|
at $v_0$ and at $v_1$ is the same even cyclic class $(a, b, c)$.
|
|
|
|
Let $F_R, F_L$ be the two faces of $\widehat{G}'_{v,i}$ on the two
|
|
sides of $e$, with $F_R$ on the right side as one walks from $v_0$ to
|
|
$v_1$. For a vertex $v \in \{v_0, v_1\}$, the non-$e$ edge of
|
|
$\partial F_R$ at $v$ is the next-clockwise edge from $e$ around $v_0$
|
|
(since at $v_0$ the right side coincides with the clockwise next edge
|
|
from $e$) and the next-counter-clockwise edge from $e$ around $v_1$
|
|
(since at $v_1$ the orientation of $e$ is reversed, so the right side
|
|
coincides with the counter-clockwise next edge from $e$).
|
|
|
|
\emph{Case~A: $\varphi(e) = a$.} In the CW order $(a, b, c)$ at $v_0$
|
|
the next-CW edge from $e$ has colour $b$; in the same CW order
|
|
$(a, b, c)$ at $v_1$ the next-CCW edge from $e$ has colour $c$ (since
|
|
CCW-next from $a$ in cyclic order $(a, b, c)$ is $c$). Hence the
|
|
non-$e$ edge of $\partial F_R$ at $v_0$ has colour $b$, while the
|
|
non-$e$ edge of $\partial F_R$ at $v_1$ has colour $c$ --- these
|
|
differ. Symmetrically, the non-$e$ edges of $\partial F_L$ at $v_0$
|
|
and $v_1$ have colours $c$ and $b$ respectively, again different.
|
|
Hence the colour-$b$ edges at $v_0$ and $v_1$ lie on opposite faces
|
|
of $e$, and the same for the colour-$c$ edges; no face of
|
|
$\widehat{G}'_{v,i}$ contains two same-coloured non-$e$ edges at
|
|
$\{v_0, v_1\}$.
|
|
|
|
\emph{Case~B: $\varphi(e) = b$.} By the analogous reasoning, the
|
|
non-$e$ edges of $\partial F_R$ at $v_0$ and $v_1$ have colours $c$
|
|
and $a$ respectively, and those of $\partial F_L$ have colours $a$
|
|
and $c$. The colour-$a$ edges at $v_0, v_1$ thus lie on opposite
|
|
faces of $e$, and so do the colour-$c$ edges.
|
|
|
|
In either case, no face $F$ of $\widehat{G}'_{v,i}$ has two same-coloured
|
|
non-$e$ edges at $\{v_0, v_1\}$ on $\partial F$, so the clause-(3) arc
|
|
(with $e$ as the unique $\partial F$-edge between $e_1$ and $e_2$ at the
|
|
endpoints of $e$) cannot be realised.
|
|
\end{proof}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=0.98\textwidth]{fig_lemma_kempe_heawood.png}
|
|
\caption{The two cases in the proof of
|
|
Lemma~\ref{lem:kempe-heawood-constant}. Vertices $v_0, v_1$ are
|
|
consecutive on the $\{a, b\}$-Kempe cycle $K$, joined by an edge $e$,
|
|
with the lemma's hypothesis $h_\varphi(v_0) = h_\varphi(v_1) = +1$ ---
|
|
so both vertices share the clockwise colour order $(a, b, c)$.
|
|
\emph{Left (Case~A):} when $\varphi(e) = a$, the colour-$b$ edge at
|
|
$v_0$ lies south of $e$ (on $\partial F_R$) and the colour-$b$ edge at
|
|
$v_1$ lies north of $e$ (on $\partial F_L$); the two would-be witness
|
|
edges are on opposite faces, so no face of $\widehat{G}'_{v,i}$ contains
|
|
both. \emph{Right (Case~B):} when $\varphi(e) = b$, the colour-$a$
|
|
edges at $v_0, v_1$ are likewise on opposite sides of $e$. In either
|
|
case the clause-$(3)$ arc of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
cannot be realised at $e$.}
|
|
\label{fig:lemma-kempe-heawood}
|
|
\end{figure}
|
|
|
|
\begin{lemma}[If Conjecture 5.1 fails, both Kempe cycles through merged have constant Heawood number]
|
|
\label{lem:both-kempe-constant}
|
|
Let $G$, $\widehat{G}'_{v,i}$, $\varphi$ be as in
|
|
Lemma~\ref{lem:kempe-heawood-constant}, set $a := \varphi(\mathrm{merged})$,
|
|
and let $K_b, K_c$ be the two Kempe cycles of $\varphi$ through the
|
|
merged edge --- the $\{a, b\}$-Kempe cycle and the $\{a, c\}$-Kempe
|
|
cycle, where $\{b, c\} = \{1, 2, 3\} \setminus \{a\}$. If no triple
|
|
$(F, e_1, e_2)$ satisfies clauses~(1)--(3) of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} on
|
|
$(G, \widehat{G}'_{v,i}, \varphi)$, then $h_\varphi$ is constant on
|
|
$V(K_b)$ and on $V(K_c)$, and the two constants agree (so all of
|
|
$V(K_b) \cup V(K_c)$ shares a common Heawood number).
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
We prove the contrapositive: if $h_\varphi$ is non-constant on
|
|
$V(K_b)$ (the argument for $K_c$ is identical), then a triple
|
|
$(F, e_1, e_2)$ realising clauses~(1)--(3) of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
exists. The argument is precisely the case analysis of
|
|
Lemma~\ref{lem:kempe-heawood-constant} run with the opposite Heawood
|
|
hypothesis.
|
|
|
|
Let $v_0, v_1 \in V(K_b)$ be consecutive on $K_b$, joined by an edge
|
|
$e \in E(K_b)$, with $h_\varphi(v_0) \neq h_\varphi(v_1)$. After
|
|
possibly swapping take $h_\varphi(v_0) = +1$ and $h_\varphi(v_1) = -1$,
|
|
so by Definition~\ref{def:heawood-number} the clockwise cyclic colour
|
|
order at $v_0$ is the even class $(a, b, c)$ and at $v_1$ is the odd
|
|
class $(a, c, b)$.
|
|
|
|
If $\varphi(e) = a$, the next-CW edge from $e$ at $v_0$ has colour $b$,
|
|
and the next-CCW edge from $e$ at $v_1$ also has colour $b$ (since the
|
|
CCW-next from $a$ in $(a, c, b)$ is $b$). Both these $b$-edges lie on
|
|
$\partial F_R$, where $F_R$ is the face on the right of $e$ walking
|
|
$v_0 \to v_1$; $e$ is the unique $\partial F_R$-edge between them on
|
|
one arc. Setting $e_1, e_2$ to be these $b$-edges gives a triple with
|
|
$\varphi(e_1) = \varphi(e_2) = b$, both on $K_b$ along with merged,
|
|
and with neither equal to merged (which has colour $a$).
|
|
|
|
If $\varphi(e) = b$, the symmetric argument places the colour-$a$
|
|
edges at $v_0, v_1$ on $\partial F_L$ with $e$ between them; choosing
|
|
$(v_0, v_1)$ so that neither is an endpoint of merged (possible since
|
|
at most two $K_b$-vertices --- the endpoints of merged --- could
|
|
force this issue, and a non-constant $h_\varphi$ on $K_b$ guarantees a
|
|
differing-Heawood pair away from them) yields the witness.
|
|
|
|
Either way $(F, e_1, e_2)$ contradicts the hypothesis, so $h_\varphi$
|
|
must be constant on $V(K_b)$. The same argument with $K_c$ in place of
|
|
$K_b$ gives constancy on $V(K_c)$. The merged edge belongs to both
|
|
cycles, so its two endpoints --- which lie on $V(K_b) \cap V(K_c)$ ---
|
|
force the two constants to coincide.
|
|
\end{proof}
|
|
|
|
\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{Heawood1898}
|
|
P.~J.~Heawood,
|
|
\emph{On the four-colour map theorem},
|
|
Quart. J.~Pure Appl. Math. \textbf{29} (1898), 270--285.
|
|
|
|
\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}
|