dual_decomposition: abstract + intro, rename to "edge suppression", bib
- Fill in the abstract and add a Section 1 introduction. - Rename "cubic-graph edge contraction" -> "edge suppression" throughout (section, definition, theorem, captions, prose, labels). The PNG filenames keep their old paths and still resolve. - Reframe edge suppression as a classical operation we recall, not a new concept we introduce; the face-monochromatic-pair conjecture (with its strengthening) is the sole contribution. - Add a bibliography citing Appel-Haken Parts I and II, Robertson-Sanders- Seymour-Thomas, and Gonthier, and \cite them in the intro. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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@@ -45,12 +45,99 @@ graph, edge connectivity, cyclic edge cut, Tait colouring, $3$-edge-colouring}
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\dedicatory{}
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\begin{abstract}
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% TODO: 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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@@ -118,6 +205,7 @@ 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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@@ -393,61 +481,63 @@ distinct colours, contradicting Lemma~\ref{lem:chord-apex} applied to
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$\varphi'$.
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\end{proof}
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\section{Cubic-graph edge contraction}
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\section{Edge suppression}
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\label{sec:edge-suppression}
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We introduce a cubic-preserving analogue of edge contraction which turns
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out --- under planar duality --- to coincide with simple-graph contraction
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on the dual side. It will be the central tool in
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We recall the classical operation of \emph{edge suppression} on cubic
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plane graphs: delete the edge and smooth the two resulting degree-$2$
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endpoints. Under planar duality this coincides with simple-graph
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contraction on the dual side. It will be the central tool in
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Section~\ref{sec:toward-4ct} below, where we formulate a sufficient
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condition for the Four Colour Theorem.
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\begin{definition}[Cubic-graph edge contraction]
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\label{def:cubic-edge-contraction}
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Let $H$ be a cubic plane graph and $e = uv$ an edge of $H$ with $u \neq v$ and
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no edge of $H$ parallel to $e$. The \emph{cubic-graph edge contraction} of $H$
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along $e$ is the graph $H'$ obtained in two steps:
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\begin{definition}[Edge suppression]
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\label{def:edge-suppression}
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Let $H$ be a cubic plane graph and $e = uv$ an edge of $H$ with $u \neq v$
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and no edge of $H$ parallel to $e$. The \emph{edge suppression} of $H$ at
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$e$ is the graph $H'$ obtained in two steps:
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\begin{enumerate}
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\item \emph{Delete} the edge $e$; the endpoints $u$ and $v$ each drop to
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degree $2$.
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\item \emph{Smooth} each of $u$ and $v$: at $u$, replace $u$ and its two
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remaining incident edges $ua, ub$ by a single new edge $ab$; do the
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same at $v$. Both vertices $u$ and $v$ are removed, and two new edges
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are added in their place.
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remaining incident edges $ua, ub$ by a single new edge $ab$; do
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the same at $v$. Both vertices $u$ and $v$ are removed, and two
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new edges are added in their place.
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\end{enumerate}
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Provided the smoothings do not introduce a loop or parallel edge, $H'$ is
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again a cubic plane graph, with $|V(H')| = |V(H)| - 2$ and
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$|E(H')| = |E(H)| - 3$.
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Equivalently, $H'$ is the planar dual of $\mathrm{dual}(H) / e^{*}$, where
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$e^{*}$ is the edge of $\mathrm{dual}(H)$ crossing $e$ and the contraction
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on the right-hand side is simple-graph contraction (loops removed, parallel
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edges absorbed). Under planar duality, contracting $e^{*}$ in
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$\mathrm{dual}(H)$ merges the two triangular faces of $\mathrm{dual}(H)$
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incident to $e^{*}$, and the parallel-edge cleanup corresponds exactly to
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the smoothing step on the primal side.
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Equivalently, $H'$ is the planar dual of $\mathrm{dual}(H) / e^{*}$,
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where $e^{*}$ is the edge of $\mathrm{dual}(H)$ crossing $e$ and the
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contraction on the right-hand side is simple-graph contraction (loops
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removed, parallel edges absorbed). Under planar duality, contracting
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$e^{*}$ in $\mathrm{dual}(H)$ merges the two triangular faces of
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$\mathrm{dual}(H)$ incident to $e^{*}$, and the parallel-edge cleanup
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corresponds exactly to the smoothing step on the primal side.
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\end{definition}
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\begin{figure}[h]
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\centering
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\includegraphics[width=0.95\textwidth]{fig_cubic_edge_contraction.png}
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\caption{Cubic-graph edge contraction
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(Definition~\ref{def:cubic-edge-contraction}). Left: a fragment of a cubic
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plane graph with the contracted edge $e = uv$ highlighted in red. Middle:
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deleting $e$ leaves $u$ and $v$ of degree~$2$. Right: smoothing $u$ and $v$
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replaces each pair of incident edges by a single new edge, removing $u, v$
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and giving a cubic plane graph again.}
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\label{fig:cubic-edge-contraction}
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\caption{Edge suppression
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(Definition~\ref{def:edge-suppression}). Left: a fragment of a cubic
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plane graph with the suppressed edge $e = uv$ highlighted in red. Middle:
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deleting $e$ leaves $u$ and $v$ of degree~$2$. Right: smoothing $u$ and
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$v$ replaces each pair of incident edges by a single new edge, removing
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$u, v$ and giving a cubic plane graph again.}
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\label{fig:edge-suppression}
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\end{figure}
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\begin{theorem}[Cubic contraction across a 4-face preserves 3-edge-colourability]
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\label{thm:cubic-contraction-4face}
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\begin{theorem}[Edge suppression across a 4-face preserves 3-edge-colourability]
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\label{thm:edge-suppression-4face}
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Let $H$ be a cubic plane graph with a proper $3$-edge-colouring $\varphi$,
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let $f$ be a face of $H$ with $|\partial f| = 4$, and let $e_0, e_1$ be the
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two edges of $\partial f$ sharing no endpoint (the opposite pair on the
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$4$-cycle $\partial f$). If $\varphi(e_0) \neq \varphi(e_1)$ and the
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cubic-graph edge contraction of $H$ along $e_0$
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(Definition~\ref{def:cubic-edge-contraction}) is well-defined (no loops or
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parallel edges are created), then the contracted graph admits a proper
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let $f$ be a face of $H$ with $|\partial f| = 4$, and let $e_0, e_1$ be
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the two edges of $\partial f$ sharing no endpoint (the opposite pair on
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the $4$-cycle $\partial f$). If $\varphi(e_0) \neq \varphi(e_1)$ and the
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edge suppression of $H$ at $e_0$
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(Definition~\ref{def:edge-suppression}) is well-defined (no loops or
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parallel edges are created), then the suppressed graph admits a proper
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$3$-edge-colouring.
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\end{theorem}
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@@ -466,8 +556,8 @@ $\varphi(w_1) = b$ and $\varphi(w_2) = a$. Symmetrically $\varphi(e_3) = c$,
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$\varphi(w_0) = b$, and $\varphi(w_3) = a$. In particular
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$\varphi(w_0) = \varphi(w_1) = b$.
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\emph{Construction of $\varphi'$.} Let $H'$ denote the cubic-graph edge
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contraction of $H$ along $e_0$; its new edges are $e_3' := v_3 u_0$
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\emph{Construction of $\varphi'$.} Let $H'$ denote the edge suppression
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of $H$ at $e_0$; its new edges are $e_3' := v_3 u_0$
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(replacing $e_3$ and $w_0$ via the smoothing at $v_0$) and
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$e_2' := v_2 u_1$ (replacing $e_2$ and $w_1$ via the smoothing at $v_1$).
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Define $\varphi' \colon E(H') \to \{1, 2, 3\}$ by
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@@ -506,16 +596,16 @@ $\varphi'$ is proper.
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\centering
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\includegraphics[width=0.98\textwidth]{fig_thm_cubic_contraction_4face.png}
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\caption{The recolouring used in the proof of
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Theorem~\ref{thm:cubic-contraction-4face}. Left: the $4$-face $f$ of $H$
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Theorem~\ref{thm:edge-suppression-4face}. Left: the $4$-face $f$ of $H$
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under $\varphi$, with the forced colours $\varphi(e_0) = a$,
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$\varphi(e_1) = b$, $\varphi(e_2) = \varphi(e_3) = c$,
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$\varphi(w_0) = \varphi(w_1) = b$, and $\varphi(w_2) = \varphi(w_3) = a$.
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Right: the contracted graph $H'$ under $\varphi'$. The smoothed-in edges
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Right: the suppressed graph $H'$ under $\varphi'$. The smoothed-in edges
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$e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is
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recoloured from $b$ to $c$; every edge outside the face neighbourhood
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keeps its $\varphi$-colour (dotted in red: the five edges of $H$ removed
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by the contraction).}
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\label{fig:thm-cubic-contraction-4face}
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by the suppression).}
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\label{fig:thm-edge-suppression-4face}
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\end{figure}
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\section{The face-monochromatic-pair conjecture and the Four Colour Theorem}
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@@ -524,14 +614,7 @@ by the contraction).}
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The following conjecture identifies a structural property of every proper
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$3$-edge-colouring of a reduced dual of a minimal counterexample. If true,
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it implies the Four Colour Theorem via
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Theorem~\ref{thm:cubic-contraction-4face}.
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The reduced-dual construction of Definition~\ref{def:reduced-dual} can also
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be applied \emph{iteratively}, producing a sequence $H_1, H_2, \dots$ of
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progressively smaller cubic plane graphs and tracking an accumulating set
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of ``protected'' edges; that iterated reduction, with its termination
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analysis and structural lemmas about spike/merged pairs at each stage, is
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the subject of a companion paper and is not used below.
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Theorem~\ref{thm:edge-suppression-4face}.
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\begin{conjecture}[Face-monochromatic-pair conjecture]
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\label{conj:face-monochromatic-pair-on-merged-kempe-cycle}
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@@ -601,7 +684,7 @@ has no content there.
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The following strengthening adds a fourth clause that arranges the new
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$4$-face $f_n$ to satisfy the hypotheses of
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Theorem~\ref{thm:cubic-contraction-4face}.
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Theorem~\ref{thm:edge-suppression-4face}.
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\begin{conjecture}[Strengthening of
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Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}]
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@@ -674,11 +757,11 @@ carry distinct $\varphi'$-colours, the cyclic colour pattern on
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$\partial f_n$ must be $(c, a, c, b)$ up to rotation and relabelling, with
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the two $c$-edges opposite and the two remaining opposite edges carrying
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the distinct colours $a$ and $b$. Those two opposite edges therefore
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satisfy the hypothesis of Theorem~\ref{thm:cubic-contraction-4face}: they
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satisfy the hypothesis of Theorem~\ref{thm:edge-suppression-4face}: they
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lie on the $4$-face $f_n$, share no endpoint, and have different
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$\varphi'$-colours. Theorem~\ref{thm:cubic-contraction-4face} then produces
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a proper $3$-edge-colouring of the cubic-graph edge contraction
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$\widehat{G}'^{+} \!\setminus\! e$ along the $a$-coloured one.
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$\varphi'$-colours. Theorem~\ref{thm:edge-suppression-4face} then produces
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a proper $3$-edge-colouring of the edge suppression of
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$\widehat{G}'^{+}$ at the $a$-coloured one.
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Case~(ii) of clause~(4) is conjecturally reducible to case~(i) by a single
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Kempe swap on the $\{b, c\}$-cycle through $X_1 X_2$: by hypothesis that
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@@ -688,9 +771,9 @@ $\partial f_n$ unchanged --- placing $\partial f_n$ into the three-colour
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pattern of case~(i).
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\smallskip\noindent\emph{Consequence.}
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Theorem~\ref{thm:cubic-contraction-4face} now produces a proper
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$3$-edge-colouring of the cubic-graph edge contraction of
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$\widehat{G}'^{+}$ along the chosen edge of $f_n$. Combined with the
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Theorem~\ref{thm:edge-suppression-4face} now produces a proper
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$3$-edge-colouring of the edge suppression of $\widehat{G}'^{+}$ at the
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chosen edge of $f_n$. Combined with the
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chord-apex and Kempe-cycle structure of $\widehat{G}'_{v,i}$
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(Lemmas~\ref{lem:chord-apex} and~\ref{lem:kempe-spike}), this yields a
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proper $3$-edge-colouring of $G'$, and by Tait's correspondence a proper
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@@ -700,4 +783,26 @@ Conjecture~\ref{conj:face-monochromatic-pair-strengthened} implies the
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Four Colour Theorem.
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\end{remark}
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\begin{thebibliography}{9}
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\bibitem{AH77a}
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K.~Appel and W.~Haken,
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\emph{Every planar map is four colorable. Part~I: Discharging},
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Illinois J.~Math. \textbf{21} (1977), 429--490.
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\bibitem{AHK77}
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K.~Appel, W.~Haken, and J.~Koch,
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\emph{Every planar map is four colorable. Part~II: Reducibility},
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Illinois J.~Math. \textbf{21} (1977), 491--567.
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\bibitem{RSST97}
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N.~Robertson, D.~P.~Sanders, P.~Seymour, and R.~Thomas,
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\emph{The four-colour theorem},
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J.~Combin. Theory Ser.~B \textbf{70} (1997), 2--44.
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\bibitem{Gonthier08}
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G.~Gonthier,
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\emph{Formal proof---the four-color theorem},
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Notices Amer. Math. Soc. \textbf{55} (2008), no.~11, 1382--1393.
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\end{thebibliography}
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\end{document}
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