didericis
36be3e14cb
Per user's request: try option 3 (push case analysis) as a sanity
check on the G'-pentagon fallback conjecture. Result: trivial
pigeonhole closes the |S| ≤ 1 case but no further.
Added:
- Lemma (Partial proof, |S| ≤ 1): a single uncovered vertex hits
at most 3 of the ≥ 6 G'-pentagons of the reduced dual, so 3
pentagons remain fully covered. Trivial pigeonhole.
- Remark "The pigeonhole stops at |S| = 1; the proof begins to
resemble discharging": Theorem deciding-face-partial-extended
+ Lemma gprime-pigeonhole cover ~91% of chord-apex+Kempe
configurations (73.87% with |S|=0 + 17% with |S|=1). The
remaining ~9% have |S| ≥ 2 and need finer graph-structural
input, exactly the discharging flavour Appel-Haken / RSST /
Gonthier used for 4CT. We stop the case-by-case route here.
So the sanity check confirms the user's intuition: continuing the
case analysis would replay the irreducible-configurations approach
in different vocabulary. Pivoting to option 2 (global argument)
next.
Paper stays at 22 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
1644 lines
75 KiB
TeX
1644 lines
75 KiB
TeX
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\usepackage{graphicx}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{corollary}[theorem]{Corollary}
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\newtheorem{proposition}[theorem]{Proposition}
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\theoremstyle{definition}
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\newtheorem{example}[theorem]{Example}
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\newtheorem{algorithm}[theorem]{Algorithm}
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\theoremstyle{remark}
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\numberwithin{equation}{section}
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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
|
|
pentagonal face of $G'$ and rewire its five external vertices around a
|
|
new apex vertex and a chord; the resulting \emph{reduced dual}
|
|
$\widehat{G}'_{v,i}$ 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
|
|
face-monochromatic-pair conjecture, in its strengthened form, asserts the
|
|
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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|
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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
|
|
by computer-assisted case analysis on a discharging argument over a finite
|
|
set of unavoidable reducible configurations. Appel and Haken's original
|
|
proof~\cite{AH77a, AHK77}, the Robertson--Sanders--Seymour--Thomas
|
|
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 $G'$,
|
|
a cubic plane graph whose proper $3$-edge-colourings correspond by Tait's
|
|
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
|
|
rewire its five external vertices, obtaining a smaller cubic plane graph
|
|
$\widehat{G}'_{v,i}$ --- the \emph{reduced dual} --- which by minimality
|
|
\emph{is} properly $3$-edge-colourable. Two structural lemmas constrain
|
|
every such colouring: a \emph{chord-apex} condition (Lemma~\ref{lem:chord-apex})
|
|
forcing two named edges to share a colour, and a pair of
|
|
\emph{Kempe-cycle} conditions (Lemma~\ref{lem:kempe-spike}) placing four
|
|
of the rewired edges on common bichromatic Kempe cycles. These constraints
|
|
are the starting point of the present development.
|
|
|
|
The main contribution of the paper is the
|
|
\emph{face-monochromatic-pair conjecture}
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|
(Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}) and
|
|
its strengthening
|
|
(Conjecture~\ref{conj:face-monochromatic-pair-strengthened}), which we
|
|
show together imply the Four Colour Theorem. The supporting ingredients
|
|
are the chord-apex and Kempe-cycle lemmas on reduced-dual colourings, the
|
|
classical operation of \emph{edge suppression} (delete the edge and smooth
|
|
its two degree-$2$ endpoints; equivalently, simple-graph contraction in
|
|
the dual triangulation; recalled in Section~\ref{sec:edge-suppression}),
|
|
and an observation that suppression preserves $3$-edge-colourability when
|
|
applied across a $4$-face whose two opposite boundary edges carry
|
|
different colours (Theorem~\ref{thm:edge-suppression-4face}).
|
|
|
|
The strategy is to construct, from a putative minimal counterexample's
|
|
reduced-dual colouring, a $4$-face $f_n$ in a slightly modified graph
|
|
$\widehat{G}'^{+}$ to which the suppression theorem applies; the
|
|
suppression then produces a properly $3$-edge-coloured graph from which a
|
|
$3$-edge-colouring of $G'$ can be recovered, contradicting the
|
|
non-$4$-colourability of $G$. The face-monochromatic-pair conjecture
|
|
asserts the existence of the structural data ($F, e_1, e_2$) needed to
|
|
build $f_n$; the strengthening guarantees that $f_n$'s boundary colouring
|
|
falls under the suppression theorem's hypothesis. Both conjectures have
|
|
been verified computationally on all chord-apex+Kempe colourings of
|
|
reduced duals up to $|V(G)| \leq 20$, with the weaker form going up to
|
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$|V(G)| \leq 21$.
|
|
|
|
\paragraph{Organization.} Section~\ref{sec:minimal} fixes the
|
|
minimal-counterexample framework: $G$ is a triangulation,
|
|
$\delta(G) \geq 5$, and every triangulation on fewer vertices is properly
|
|
$4$-colourable. Section~\ref{sec:reduced-dual} introduces the reduced dual
|
|
$\widehat{G}'_{v,i}$ and proves the chord-apex and Kempe-cycle lemmas.
|
|
Section~\ref{sec:edge-suppression} defines edge suppression and proves
|
|
its $4$-face $3$-edge-colourability theorem.
|
|
Section~\ref{sec:toward-4ct} states the two conjectures, reports the
|
|
empirical verification, and gives the implication to 4CT.
|
|
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|
\paragraph{Companion paper.} An iterated version of the reduced-dual
|
|
construction --- producing a sequence $H_1, H_2, \dots$ of progressively
|
|
smaller cubic plane graphs and tracking an accumulating ``protected'' edge
|
|
set --- is the subject of a companion paper. The present paper does not
|
|
use that iteration.
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\section{The minimal counterexample}
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|
\label{sec:minimal}
|
|
|
|
Throughout, a \emph{triangulation} is a simple plane graph, with a fixed
|
|
embedding, in which every face --- including the outer face --- is bounded by a
|
|
triangle. We first reduce to triangulations, then record the degree properties a
|
|
smallest counterexample must have.
|
|
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|
\begin{lemma}[Reduction to triangulations]
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|
\label{lem:triangulate}
|
|
If every triangulation is properly $4$-vertex-colourable, then so is every plane
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|
graph.
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|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
Let $H$ be a plane graph. Add edges to $H$, maintaining planarity, until no
|
|
further edge can be added; the result is a triangulation $H^{+}$ on the same
|
|
vertex set with $E(H) \subseteq E(H^{+})$. A proper $4$-colouring of $H^{+}$
|
|
restricts to a proper $4$-colouring of $H$, since every edge of $H$ is an edge of
|
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$H^{+}$.
|
|
\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
|
|
$4$-vertex-colouring. We call $G$ a \emph{minimal counterexample}. By minimality,
|
|
every triangulation on fewer than $|V(G)|$ vertices is properly
|
|
$4$-colourable.
|
|
\end{definition}
|
|
|
|
\begin{remark}
|
|
Since every triangulation on at most four vertices is properly $4$-colourable
|
|
(the largest being $K_4$), a minimal counterexample has $|V(G)| \ge 5$; the degree
|
|
bound below sharpens this to $|V(G)| \ge 12$.
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|
\end{remark}
|
|
|
|
\begin{lemma}[Minimum degree]
|
|
\label{lem:mindeg}
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|
A minimal counterexample $G$ has minimum degree $\delta(G) \ge 5$.
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
Suppose some vertex $v$ has $\deg(v) = d \le 4$.
|
|
|
|
If $d \le 3$, let $G' = G - v$. Then $G'$ is a plane graph on fewer vertices, so
|
|
by Definition~\ref{def:minimal} and Lemma~\ref{lem:triangulate} it has a proper
|
|
$4$-colouring. The at most three neighbours of $v$ use at most three colours, so
|
|
a fourth colour is free for $v$, extending the colouring to $G$ --- a
|
|
contradiction.
|
|
|
|
If $d = 4$, again $4$-colour $G - v$. If the four neighbours of $v$ use at most
|
|
three colours we extend as before, so assume they receive all four colours; let
|
|
$v_1, v_2, v_3, v_4$ be the neighbours in cyclic order around $v$, coloured
|
|
$1,2,3,4$. Consider the subgraph induced by the colour classes $1$ and $3$, and
|
|
let $K$ be its connected component containing $v_1$. If $v_3 \notin K$, swap
|
|
colours $1$ and $3$ on $K$; now no neighbour of $v$ is coloured $1$, freeing it
|
|
for $v$. If $v_3 \in K$, then a $1$--$3$ Kempe chain joins $v_1$ to $v_3$, and
|
|
this chain together with $v$ encloses exactly one of $v_2, v_4$; hence the
|
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$2$--$4$ component containing $v_2$ cannot also reach $v_4$, and swapping colours
|
|
$2$ and $4$ on it frees colour $2$ for $v$. Either way the colouring extends to
|
|
$G$, a contradiction.
|
|
|
|
Hence $\delta(G) \ge 5$.
|
|
\end{proof}
|
|
|
|
\section{The reduced dual}
|
|
\label{sec:reduced-dual}
|
|
|
|
Write $G'$ for the dual of $G$: since $G$ is a triangulation, $G'$ is a cubic
|
|
plane graph in which each vertex of $G$ corresponds to a face of $G'$, each face
|
|
of $G$ to a vertex of $G'$, and each edge to a dual edge. A vertex of $G$ of
|
|
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
|
|
plane graph will be useful in Section~\ref{sec:toward-4ct}.
|
|
|
|
\begin{definition}[Heawood number of a vertex]
|
|
\label{def:heawood-number}
|
|
Let $H$ be a cubic plane graph with a fixed planar embedding, and let
|
|
$\varphi \colon E(H) \to \{1, 2, 3\}$ be a proper $3$-edge-colouring. At
|
|
each vertex $v \in V(H)$, the three incident edges receive three distinct
|
|
colours; reading them in clockwise order around $v$ gives a cyclic
|
|
permutation of $(1, 2, 3)$. The \emph{Heawood number} of $v$ is
|
|
\[
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|
h_\varphi(v) :=
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|
\begin{cases}
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+1 & \text{if the clockwise cyclic colour order at $v$ is }(1, 2, 3), \\
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|
-1 & \text{if it is }(1, 3, 2).
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|
\end{cases}
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|
\]
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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.
|
|
The labels are due to Heawood~\cite{Heawood1898}, who introduced them as
|
|
part of his analysis of $3$-edge-colourings of cubic plane graphs.
|
|
\end{definition}
|
|
|
|
By Lemma~\ref{lem:mindeg}, $\delta(G) \ge 5$, and Euler's formula gives
|
|
$\sum_{u \in V(G)}(6 - \deg u) = 12$, so $G$ has a vertex of degree exactly $5$
|
|
(indeed at least twelve). Fix such a vertex $v$. Its dual face $F_v$ is a
|
|
pentagon, bounded by the five dual vertices corresponding to the five faces of
|
|
$G$ incident to $v$.
|
|
|
|
\begin{definition}[Reduced dual]
|
|
\label{def:reduced-dual}
|
|
Let $v$ be a degree-$5$ vertex of $G$ with pentagonal dual face $F_v$, and fix an
|
|
index $i \in \{0,1,2,3,4\}$. The \emph{reduced dual} $\widehat{G}'_{v,i}$ is the
|
|
plane graph obtained from $G'$ as follows.
|
|
\begin{enumerate}
|
|
\item Delete the five dual vertices on the boundary of $F_v$, together with all
|
|
edges incident to them. Each deleted vertex is cubic, with two edges on
|
|
$\partial F_v$ and one edge leaving $F_v$; deleting the five boundary
|
|
vertices therefore removes the five external edges as well, dropping their
|
|
five outer endpoints from degree $3$ to degree $2$. These five degree-$2$
|
|
vertices lie on the boundary of a single face $F$ of the resulting graph.
|
|
\item List the five degree-$2$ vertices in clockwise order around $F$ as
|
|
$A = (A_0, A_1, A_2, A_3, A_4)$.
|
|
\item Add a new vertex $v_n$ and join it to $A_i$, $A_{i+1}$, and $A_{i+2}$
|
|
(indices mod $5$) by three new edges.
|
|
\item Add a new edge between $A_{i+3}$ and $A_{i+4}$ (indices mod $5$).
|
|
\end{enumerate}
|
|
\end{definition}
|
|
|
|
\begin{remark}
|
|
Steps (3) and (4) restore cubicity: $A_i, A_{i+1}, A_{i+2}$ each gain one edge to
|
|
$v_n$ and $A_{i+3}, A_{i+4}$ each gain the new edge, so all five return to degree
|
|
$3$, and $v_n$ has degree $3$. Since $A_i,\dots,A_{i+2}$ and $A_{i+3}, A_{i+4}$
|
|
are each consecutive along $\partial F$, the new vertex and edge can be drawn
|
|
inside $F$ without crossings, so $\widehat{G}'_{v,i}$ is again a cubic plane
|
|
graph. The construction depends on the choice of $i$ up to the rotational
|
|
symmetry of $A$.
|
|
\end{remark}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step1.png}\hfill
|
|
\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step2.png}\\[0.5em]
|
|
\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step3.png}\hfill
|
|
\includegraphics[width=0.48\textwidth]{fig_reduced_dual_step4.png}
|
|
\caption{The four steps of Definition~\ref{def:reduced-dual}, illustrated on
|
|
$G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner
|
|
pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$,
|
|
leaving five degree-$2$ vertices on a new face $F$. Top right: order them
|
|
clockwise as $A_0,\dots,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$.
|
|
Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph
|
|
$\widehat{G}'_{v,0}$.}
|
|
\label{fig:reduced-dual-steps}
|
|
\end{figure}
|
|
|
|
\begin{definition}[Edges of the reduced dual]
|
|
\label{def:edge-names}
|
|
The four edges added in steps (3) and (4) of Definition~\ref{def:reduced-dual}
|
|
are named as follows. The chord $A_{i+3}A_{i+4}$ is the \emph{merged edge};
|
|
the edge $A_{i+1}v_n$ is the \emph{spike edge}; the edge $A_iv_n$ is the
|
|
\emph{side-$0$ edge}; and the edge $A_{i+2}v_n$ is the \emph{side-$1$ edge}.
|
|
In the $i = 0$ case of Figure~\ref{fig:reduced-dual-steps} these are
|
|
$\{A_3, A_4\}$, $\{A_1, v_n\}$, $\{A_0, v_n\}$, and $\{A_2, v_n\}$
|
|
respectively.
|
|
\end{definition}
|
|
|
|
We will use the following structural fact about proper $3$-edge-colourings near
|
|
a pentagonal face of a cubic plane graph; it is stated for a generic such graph
|
|
$H$, not specifically for the reduced dual.
|
|
|
|
\begin{lemma}[Pentagonal externals]
|
|
\label{lem:pentagonal-externals}
|
|
Let $H$ be a cubic plane graph and $F$ a pentagonal face of $H$, with
|
|
$\partial F$ traversed clockwise as $u_0, u_1, u_2, u_3, u_4$. For each $i$
|
|
let $f_i$ be the unique edge of $H$ incident to $u_i$ that does not lie on
|
|
$\partial F$. An assignment $\varphi$ of colours from $\{1, 2, 3\}$ to the ten
|
|
edges incident to $\{u_0, \dots, u_4\}$ is proper at every $u_i$ if and only if
|
|
there is some index $j$ such that
|
|
\[
|
|
\varphi(f_j) = \varphi(f_{j+1}) = \varphi(f_{j+2})
|
|
\quad\text{and}\quad
|
|
\{\varphi(f_{j+3}), \varphi(f_{j+4})\}
|
|
= \{1, 2, 3\} \setminus \{\varphi(f_j)\},
|
|
\]
|
|
indices mod $5$.
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
Write $e_i = u_i u_{i+1}$ for the boundary edges of $\partial F$ (indices mod
|
|
$5$). A colouring $\varphi$ is proper at every $u_i$ if and only if at each
|
|
$u_i$ the three incident edges $e_{i-1}, e_i, f_i$ receive three distinct
|
|
colours; whenever this holds, $\varphi(f_i)$ is forced to be the unique colour
|
|
in $\{1, 2, 3\} \setminus \{\varphi(e_{i-1}), \varphi(e_i)\}$, and $\varphi$
|
|
restricts to a proper $3$-edge-colouring of the cycle $\partial F$.
|
|
|
|
\textbf{($\Rightarrow$)} The line graph of $\partial F$ is $C_5$, whose
|
|
maximum independent set has size $2$, so no colour appears more than twice on
|
|
$\partial F$; and since $\partial F$ is an odd cycle, all three colours appear.
|
|
The colour multiset on $(\varphi(e_0), \dots, \varphi(e_4))$ is therefore
|
|
$(2, 2, 1)$, with the singleton at a unique position. Cyclically shifting
|
|
indices we may place this position at $0$; let $c$ be the singleton colour.
|
|
The remaining four edges form the path $e_1 e_2 e_3 e_4$, which by propriety
|
|
alternates between the other two colours, so for some labelling
|
|
$\{a, b, c\} = \{1, 2, 3\}$,
|
|
\[
|
|
(\varphi(e_0), \varphi(e_1), \varphi(e_2), \varphi(e_3), \varphi(e_4))
|
|
= (c, a, b, a, b).
|
|
\]
|
|
Reading off the forced values of $\varphi(f_i)$,
|
|
\[
|
|
\varphi(f_0) = a, \quad
|
|
\varphi(f_1) = b, \quad
|
|
\varphi(f_2) = \varphi(f_3) = \varphi(f_4) = c,
|
|
\]
|
|
which is the lemma's pattern at $j = 2$ (the cyclic shift maps this back to
|
|
the corresponding $j$ in the original indexing). This case is the unique
|
|
proper $3$-edge-colouring of $\partial F$ up to cyclic shift and permutation of
|
|
$\{1, 2, 3\}$ (since $5 \cdot 3! = 30 = P(C_5, 3)$, the chromatic polynomial of
|
|
$C_5$ at $3$), so it exhausts every proper $\varphi$.
|
|
|
|
\textbf{($\Leftarrow$)} The lemma's hypothesis is invariant under cyclic
|
|
shifts of indices and under permutations of $\{1, 2, 3\}$, so we may assume
|
|
$j = 2$, $\varphi(f_2) = \varphi(f_3) = \varphi(f_4) = c$, $\varphi(f_0) = a$,
|
|
and $\varphi(f_1) = b$, with $\{a, b, c\} = \{1, 2, 3\}$. Propriety at $u_i$
|
|
and $u_{i+1}$ requires $\varphi(e_i) \notin \{\varphi(f_i), \varphi(f_{i+1})\}$,
|
|
which gives
|
|
\[
|
|
\varphi(e_0) = c, \quad
|
|
\varphi(e_1) = a, \quad
|
|
\varphi(e_2) \in \{a, b\}, \quad
|
|
\varphi(e_3) \in \{a, b\}, \quad
|
|
\varphi(e_4) = b.
|
|
\]
|
|
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{corollary}[Per-cycle form]
|
|
\label{cor:single-cycle-non-constancy}
|
|
Let $G$, $\widehat{G}'_{v,i}$, $\varphi$ be as in
|
|
Lemma~\ref{lem:both-kempe-constant}, and let $K$ be either of the two
|
|
Kempe cycles of $\varphi$ through the merged edge. If $h_\varphi$ is not
|
|
constant on $V(K)$, then a triple $(F, e_1, e_2)$ satisfying
|
|
clauses~(1)--(3) of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} on
|
|
$(G, \widehat{G}'_{v,i}, \varphi)$ exists.
|
|
\end{corollary}
|
|
|
|
\begin{proof}
|
|
This is precisely the case analysis used to prove
|
|
Lemma~\ref{lem:both-kempe-constant}: applied to any consecutive pair of
|
|
vertices on $K$ with differing Heawood numbers, the construction in
|
|
that proof produces a clauses-(1)--(3) witness without ever needing to
|
|
inspect the other Kempe cycle.
|
|
\end{proof}
|
|
|
|
\begin{conjecture}[Constant Heawood on two edge-sharing Kempe cycles, large-face cubic plane graphs --- \textbf{FALSE}]
|
|
\label{conj:no-two-constant-kempe-cycles}
|
|
Let $H$ be a cubic plane graph in which every face has length at
|
|
least $5$, with a proper $3$-edge-colouring $\varphi$. Fix a colour
|
|
$a \in \{1, 2, 3\}$ and let $\{b, c\} = \{1, 2, 3\} \setminus \{a\}$.
|
|
Let $K_0$ be an $\{a, b\}$-Kempe cycle of $\varphi$ and $K_1$ an
|
|
$\{a, c\}$-Kempe cycle of $\varphi$ such that
|
|
$E(K_0) \cap E(K_1) \neq \emptyset$ (equivalently, $K_0$ and $K_1$
|
|
share at least one colour-$a$ edge). If $h_\varphi$ is constant on
|
|
$V(K_0)$, then $h_\varphi$ is \emph{not} constant on $V(K_1)$.
|
|
\end{conjecture}
|
|
|
|
\begin{remark}[Disproof of
|
|
Conjecture~\ref{conj:no-two-constant-kempe-cycles}]
|
|
\label{rem:no-two-constant-kempe-cycles-counterexample}
|
|
Conjecture~\ref{conj:no-two-constant-kempe-cycles} is \emph{false}.
|
|
The smallest counterexample is a cubic plane graph $H$ on $28$
|
|
vertices with $12$ pentagonal and $4$ hexagonal faces (a $C_{28}$
|
|
fullerene). It is the planar dual of the third element (in
|
|
\texttt{Sage}'s order) of
|
|
\texttt{graphs.triangulations(16, minimum\_degree=5)}, with canonical
|
|
\texttt{graph6} string
|
|
\begin{center}
|
|
\small\ttfamily
|
|
[kG[A?\_A?\_?\_?K?D?@\_CO?o?@\_??A??@C??O??AG?C????`???a???W???A\_???F.
|
|
\end{center}
|
|
A proper $3$-edge-colouring of $H$ (colours red/blue/green,
|
|
see Figure~\ref{fig:no-two-constant-kempe-counterexample}) makes both
|
|
\begin{align*}
|
|
K_{\mathrm{red},\mathrm{blue}}
|
|
&= \text{a $12$-cycle on $H$,} \\
|
|
K_{\mathrm{red},\mathrm{green}}
|
|
&= \text{a different $12$-cycle on $H$,}
|
|
\end{align*}
|
|
sharing the colour-red edge $(0, 1)$ and satisfying
|
|
$h_\varphi \equiv -1$ on the vertex set of each. Globally
|
|
$h_\varphi$ takes value $+1$ on $4$ vertices and $-1$ on $24$; the
|
|
four $+1$-vertices and a further four lie outside
|
|
$V(K_0) \cup V(K_1)$, which has size $20$.
|
|
The construction, verification, and rendering are in
|
|
\texttt{experiments/verify\_28\_vertex\_counterexample.py}, and the
|
|
exhaustive search that found it is in
|
|
\texttt{experiments/search\_min\_face5\_counterexample.py}.
|
|
|
|
\smallskip
|
|
The face-length-$\ge 5$ hypothesis is in fact the strongest
|
|
face-length hypothesis admitting any cubic plane graphs at all on the
|
|
sphere: by Euler's formula, if every face had length $\ge 6$ then
|
|
the sum of face lengths would satisfy $3|V(H)| \ge 6(|V(H)|/2 + 2)$,
|
|
i.e.\ $0 \ge 12$, contradiction. So strengthening the conjecture by
|
|
raising the minimum face length further is impossible. Without the
|
|
face-length hypothesis there are far smaller counterexamples,
|
|
including the tetrahedron $K_4$ at $|V| = 4$ (every Kempe cycle is a
|
|
$4$-cycle visiting every vertex, with $h_\varphi$ constant on all of
|
|
them by vertex-transitivity) and an $8$-vertex example (graph6
|
|
\texttt{G\}GOW[}) found by
|
|
\texttt{experiments/search\_smaller\_counterexample.py}; both have
|
|
girth $3$. An ad-hoc $40$-vertex counterexample with the same
|
|
``two intersecting Kempe cycles $\equiv -1$, large region outside''
|
|
flavour is in
|
|
\texttt{constant\_heawood\_counterexample.tikz}.
|
|
\end{remark}
|
|
|
|
\begin{figure}[h]
|
|
\centering
|
|
\includegraphics[width=0.7\textwidth]{figures/min-face-5-counterexample.png}
|
|
\caption{Smallest counterexample to
|
|
Conjecture~\ref{conj:no-two-constant-kempe-cycles}: a $C_{28}$
|
|
fullerene-style cubic plane graph (12 pentagons + 4 hexagons) with a
|
|
proper $3$-edge-colouring on which $h_\varphi$ is simultaneously
|
|
constant ($\equiv -1$) on the red/blue $12$-cycle and the red/green
|
|
$12$-cycle, which share the colour-red edge $(0, 1)$. Light-shaded
|
|
nodes are on $V(K_0) \cap V(K_1)$; medium-shaded on
|
|
$V(K_0) \cup V(K_1) \setminus V(K_0) \cap V(K_1)$; grey on neither.}
|
|
\label{fig:no-two-constant-kempe-counterexample}
|
|
\end{figure}
|
|
|
|
\subsection*{A reduction of Conjecture
|
|
\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Heawood's
|
|
face-sum identity}
|
|
|
|
The empirical work of Section~\ref{sec:reduced-dual} (the
|
|
$0/142{,}812$ result on chord-apex+Kempe colourings, recorded in
|
|
Remark~\ref{rem:heawood-empirical}) suggests a structural proof
|
|
strategy via the classical Heawood face-sum identity
|
|
\cite{Heawood1898}:
|
|
\begin{equation}
|
|
\label{eq:heawood-face-sum}
|
|
\sum_{v \in \partial f} h_\varphi(v) \;\equiv\; 0 \pmod 3
|
|
\qquad\text{for every face $f$ of $H$,}
|
|
\end{equation}
|
|
which holds for any proper $3$-edge-colouring $\varphi$ of any cubic
|
|
plane graph $H$.
|
|
|
|
\begin{conjecture}[Deciding face]
|
|
\label{conj:deciding-face}
|
|
Let $G$ be a minimal counterexample to the Four Colour Theorem, let
|
|
$\widehat{G}'_{v,i}$ be a reduced dual of $G'$, and let $\varphi$ be
|
|
a chord-apex+Kempe colouring of $\widehat{G}'_{v,i}$. Write $K_b$ and
|
|
$K_c$ for the two Kempe cycles through the spike edge
|
|
(Lemma~\ref{lem:kempe-spike}). Then $\widehat{G}'_{v,i}$ has a face
|
|
$f$ satisfying
|
|
\[
|
|
\partial f \subseteq V(K_b) \cup V(K_c)
|
|
\qquad\text{and}\qquad
|
|
|f| \not\equiv 0 \pmod 3.
|
|
\]
|
|
\end{conjecture}
|
|
|
|
\begin{theorem}
|
|
\label{thm:deciding-face-implies-conj-5-1}
|
|
Conjecture~\ref{conj:deciding-face} implies
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}.
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
We argue by contradiction. Suppose
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
fails: there exist a minimal counterexample $G$, a reduced dual
|
|
$\widehat{G}'_{v,i}$, and a chord-apex+Kempe colouring $\varphi$ of
|
|
$\widehat{G}'_{v,i}$ admitting no clauses-(1)-(3) witness of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}.
|
|
|
|
By Lemma~\ref{lem:both-kempe-constant}, the absence of any
|
|
clauses-(1)-(3) witness forces $h_\varphi$ to be constant on
|
|
$V(K_b) \cup V(K_c)$; write the common value as
|
|
$\varepsilon \in \{+1, -1\}$.
|
|
|
|
By Conjecture~\ref{conj:deciding-face}, there is a face $f$ of
|
|
$\widehat{G}'_{v,i}$ with $\partial f \subseteq V(K_b) \cup V(K_c)$
|
|
and $|f| \not\equiv 0 \pmod 3$. Applying~\eqref{eq:heawood-face-sum}
|
|
to $f$ and using $h_\varphi(v) = \varepsilon$ for every
|
|
$v \in \partial f$:
|
|
\[
|
|
\sum_{v \in \partial f} h_\varphi(v)
|
|
\;=\; \varepsilon \cdot |f|
|
|
\;\equiv\; 0 \pmod 3.
|
|
\]
|
|
Since $3$ is prime and $\gcd(|f|, 3) = 1$, we obtain
|
|
$\varepsilon \equiv 0 \pmod 3$. But $\varepsilon \in \{+1, -1\}$,
|
|
so $\varepsilon \not\equiv 0 \pmod 3$ --- contradiction.
|
|
\end{proof}
|
|
|
|
\subsection*{A partial structural proof of
|
|
Conjecture~\ref{conj:deciding-face}}
|
|
|
|
We single out a specific candidate face --- the ``flank'' face that
|
|
the spike, side-$0$ edge, and one boundary arc of the new pentagonal
|
|
hole together bound --- and prove the deciding-face property for it
|
|
when the adjacent $G'$-face is pentagonal or hexagonal. This is the
|
|
case for every reduced dual where at least one of $v$'s neighbours
|
|
$B_i$ or $B_{i+1}$ in the parent triangulation $G$ has degree $5$ or
|
|
$6$ --- e.g., for the icosahedron and the dodecahedron's other near
|
|
neighbours.
|
|
|
|
\begin{definition}[Flank face]
|
|
\label{def:flank-face}
|
|
Fix a reduced dual $\widehat{G}'_{v,i}$ and let $F$ be the post-deletion
|
|
pentagonal hole. Order the five outer endpoints
|
|
$A_0, A_1, A_2, A_3, A_4$ in clockwise order along
|
|
$\partial F$ as in Definition~\ref{def:reduced-dual}. The
|
|
\emph{lower flank face} $F_{i, i+1}^{\flat}$ of
|
|
$\widehat{G}'_{v,i}$ is the face of $\widehat{G}'_{v,i}$ whose
|
|
boundary is the closed walk
|
|
\[
|
|
v_n
|
|
\;\xrightarrow{\text{side-}0}\;
|
|
A_i
|
|
\;\xrightarrow{\partial F \text{-arc}}\;
|
|
A_{i+1}
|
|
\;\xrightarrow{\text{spike}}\;
|
|
v_n,
|
|
\]
|
|
i.e.\ the side-$0$ edge, followed by the maximal sub-arc of
|
|
$\partial F$ from $A_i$ to $A_{i+1}$ not crossing $v_n$ or the
|
|
merged edge, followed by the spike edge in reverse.
|
|
The \emph{upper flank face} $F_{i+1, i+2}^{\flat}$ is defined
|
|
analogously with side-$1$ replacing side-$0$ and $A_{i+1} \to A_{i+2}$
|
|
replacing $A_i \to A_{i+1}$.
|
|
\end{definition}
|
|
|
|
\begin{lemma}[Flank-length formula]
|
|
\label{lem:flank-length}
|
|
$|F_{i, i+1}^{\flat}| \;=\; n_i - 1$, where $n_i$ is the length of
|
|
the $G'$-face $F_i$ adjacent to $F_v$ across the edge
|
|
$(B_i, B_{i+1})$ in the pre-reduction dual $G'$. (Symmetrically,
|
|
$|F_{i+1, i+2}^{\flat}| = n_{i+1} - 1$.)
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
After step (1) of Definition~\ref{def:reduced-dual} the dual face
|
|
$F_i$, originally a closed walk of length $n_i$ around the boundary
|
|
edges $(B_i, B_{i+1})$, $(B_{i+1}, A_{i+1})$, $\partial F_i$
|
|
interior, and $(A_i, B_i)$, becomes (after deletion of
|
|
$B_i, B_{i+1}$ and their three incident edges) an open arc from
|
|
$A_{i+1}$ to $A_i$ of length $n_i - 3$ in the post-deletion
|
|
graph. The lower flank face $F_{i, i+1}^{\flat}$ adds back the
|
|
side-$0$ edge $(v_n, A_i)$ (length $1$) and the spike edge
|
|
$(v_n, A_{i+1})$ (length $1$). Its total boundary length is therefore
|
|
$(n_i - 3) + 1 + 1 = n_i - 1$.
|
|
\end{proof}
|
|
|
|
\begin{lemma}[Flank covering, base case $n_i = 5$]
|
|
\label{lem:flank-covering-base}
|
|
If $n_i = 5$ then $\partial F_{i, i+1}^{\flat} \subseteq V(K_b) \cup V(K_c)$.
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
When $n_i = 5$ the boundary of $F_{i, i+1}^{\flat}$ visits exactly four
|
|
vertices: $v_n$, $A_i$, a single intermediate $P$, and $A_{i+1}$. We
|
|
have $v_n, A_i, A_{i+1} \in V(K_b) \cup V(K_c)$ by
|
|
Lemma~\ref{lem:kempe-spike}, so it remains to show $P \in V(K_b) \cup V(K_c)$.
|
|
|
|
The vertex $A_{i+1}$ has degree $3$, with one of its three incident
|
|
edges being the spike (colour $c$) and the other two having colours
|
|
$\{c_0, c_1\}$ (one each, by properness). Lemma~\ref{lem:kempe-spike}
|
|
gives $A_{i+1} \in V(K_b) \cap V(K_c)$, so $K_b$ uses the spike
|
|
together with $A_{i+1}$'s colour-$c_0$ edge, and $K_c$ uses the spike
|
|
together with $A_{i+1}$'s colour-$c_1$ edge. The two non-spike edges
|
|
at $A_{i+1}$ are exactly $A_{i+1} P$ (the boundary edge into the lower
|
|
flank) and $A_{i+1} P'$ (the boundary edge into the upper flank,
|
|
$P' \in F_{i+1, i+2}^{\flat}$). Whichever of these is $A_{i+1} P$,
|
|
its colour is in $\{c_0, c_1\}$, so the corresponding Kempe cycle
|
|
($K_b$ if colour $c_0$, $K_c$ if colour $c_1$) walks $A_{i+1} \to P$,
|
|
placing $P$ in that cycle's vertex set.
|
|
\end{proof}
|
|
|
|
\begin{lemma}[Flank covering, $n_i = 6$; \textbf{empirically FALSE in full
|
|
generality}]
|
|
\label{lem:flank-covering-hex}
|
|
\emph{Status:} The naive statement ``if $n_i = 6$ then
|
|
$\partial F_{i, i+1}^{\flat} \subseteq V(K_b) \cup V(K_c)$'' is
|
|
\emph{false} in full generality. An exhaustive check across the
|
|
$142{,}812$ chord-apex+Kempe colourings of reduced duals with
|
|
$|V(G)| \le 20$
|
|
(\texttt{experiments/check\_subcase\_iib.py}) finds:
|
|
\begin{itemize}
|
|
\item $9{,}228$ colourings ($6.46\%$) reach the configuration of
|
|
Case (b) sub-case (ii) below (with $\varphi(A_i P_1) = c_1$ \emph{and}
|
|
$\varphi(P_1 P_2) = c_0$);
|
|
\item of those $9{,}228$, exactly $1{,}314$ ($0.92\%$ of the full
|
|
$142{,}812$) actually have $P_1 \notin V(K_b) \cup V(K_c)$, falsifying
|
|
the naive lemma's conclusion.
|
|
\end{itemize}
|
|
|
|
The argument below correctly handles Case~(a) and Case~(b)
|
|
sub-case~(i), and so establishes only the weakened conclusion
|
|
\[
|
|
\partial F_{i, i+1}^{\flat} \;\subseteq\; V(K_b) \cup V(K_c)
|
|
\qquad\text{provided either}\qquad
|
|
\begin{cases}
|
|
\varphi(A_i P_1) = c, \quad\text{or} \\
|
|
\varphi(A_i P_1) = c_1 \;\text{and}\; \varphi(P_1 P_2) = c.
|
|
\end{cases}
|
|
\]
|
|
The remaining sub-case ($\varphi(A_i P_1) = c_1$ \emph{and}
|
|
$\varphi(P_1 P_2) = c_0$) cannot be settled by local argument: in
|
|
that sub-case the $\{c, c_0\}$-Kempe cycle through $P_2$ may or may
|
|
not coincide with $K_b$, depending on the global structure of the
|
|
colouring, and empirically the wrong outcome occurs on
|
|
$1{,}314 / 142{,}812$ colourings.
|
|
|
|
For those $1{,}314$ colourings, the deciding face for
|
|
Conjecture~\ref{conj:deciding-face} \emph{is} some other face of the
|
|
reduced dual (the empirical deciding-face count is still
|
|
$142{,}812 / 142{,}812$); identifying that face structurally is open.
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
The boundary now visits five vertices: $v_n$, $A_i$, $P_1, P_2$,
|
|
$A_{i+1}$, with $P_1$ adjacent to $A_i$ and $P_2$ adjacent to
|
|
$A_{i+1}$. The named vertices and $P_2$ are in $V(K_b) \cup V(K_c)$
|
|
by Lemma~\ref{lem:flank-covering-base}'s argument applied at
|
|
$A_{i+1}$. It remains to place $P_1$.
|
|
|
|
\emph{Case (a):} $\varphi(A_i P_1) = c$. Then $K_b$'s walk uses
|
|
$A_i$'s colour-$c$ edge (along with side-$0$), so $K_b$ walks
|
|
$A_i \to P_1$ and $P_1 \in V(K_b)$.
|
|
|
|
\emph{Case (b):} $\varphi(A_i P_1) = c_1$. Then $A_i$'s colour-$c$
|
|
edge is the \emph{other} non-side-$0$ edge of $A_i$, namely $A_i Q$
|
|
where $Q$ is the intermediate on $\partial F_{i-1, i}^{\flat}$
|
|
adjacent to $A_i$. Now consider $P_2$. By the analysis above we have
|
|
$\varphi(A_{i+1} P_2) \in \{c_0, c_1\}$, and properness at $P_2$
|
|
gives $\varphi(P_2)$'s three incident edges colours $\{c_0, c, c_1\}$
|
|
in some order, with $\varphi(A_{i+1} P_2)$ accounting for one of
|
|
$\{c_0, c_1\}$. The remaining two colours, $c$ and $(c_1$ or $c_0)$,
|
|
are split between $\varphi(P_1 P_2)$ and $\varphi(P_2 O_2)$ where
|
|
$O_2$ is $P_2$'s third (outer) neighbour. Properness at $P_1$ (which
|
|
has $\varphi(A_i P_1) = c_1$ already by Case (b) hypothesis) forbids
|
|
$\varphi(P_1 P_2) = c_1$. Hence
|
|
\[
|
|
\varphi(P_1 P_2) \;\in\; \{c, c_0\}.
|
|
\]
|
|
Case~(b) splits further on $\varphi(P_1 P_2)$.
|
|
|
|
\emph{Sub-case (i):} $\varphi(P_1 P_2) = c$. Then $P_2$'s colour-$c$
|
|
edge is $P_1 P_2$. Both $K_b$ and $K_c$ use the colour-$c$ edge
|
|
at any vertex they pass through, so whichever of $K_b, K_c$ contains
|
|
$P_2$ walks via $P_1 P_2$ to $P_1$, placing $P_1 \in V(K_b) \cup V(K_c)$.
|
|
|
|
\emph{Sub-case (ii):} $\varphi(P_1 P_2) = c_0$. Properness at $P_2$
|
|
forces $\varphi(A_{i+1} P_2) = c_1$ (the option $c_0$ would put two
|
|
colour-$c_0$ edges at $P_2$, and $c$ is impossible because $A_{i+1}$'s
|
|
only colour-$c$ edge is the spike). Hence $P_2 \in V(K_c)$ via $K_c$'s
|
|
walk $A_{i+1} \xrightarrow{c_1} P_2$. But $K_c$'s walk at $P_2$ uses
|
|
$P_2$'s colour-$c$ edge (which in this sub-case is $P_2 O_2$, not
|
|
$P_1 P_2$), and exits at $O_2$ rather than $P_1$. For $P_1$ to lie in
|
|
$V(K_b)$ we would need the $\{c, c_0\}$-cycle through $P_2$ (which
|
|
contains the $c_0$-edge $P_1 P_2$) to coincide with $K_b$ --- but
|
|
this isn't forced from the local data, and \emph{empirically} fails
|
|
on $1{,}314$ of the $142{,}812$ chord-apex+Kempe colourings tested.
|
|
|
|
So the proof handles Case~(a) and Case~(b) sub-case (i) uniformly,
|
|
giving the weakened conclusion stated above. Case~(b) sub-case (ii)
|
|
is not settled by local argument.
|
|
\end{proof}
|
|
|
|
\begin{theorem}[Partial proof of
|
|
Conjecture~\ref{conj:deciding-face} via flank face]
|
|
\label{thm:deciding-face-partial}
|
|
If $n_i = 5$ or $n_{i+1} = 5$ for the reduction index $i$, then
|
|
$\widehat{G}'_{v,i}$ has a deciding face: one of the two flank faces
|
|
$F_{i, i+1}^{\flat}$ or $F_{i+1, i+2}^{\flat}$.
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
Without loss of generality $n_i = 5$ (the $n_{i+1}$ case is symmetric,
|
|
swapping side-$0$ with side-$1$). By Lemma~\ref{lem:flank-length},
|
|
$|F_{i, i+1}^{\flat}| = 4 \not\equiv 0 \pmod 3$. By
|
|
Lemma~\ref{lem:flank-covering-base},
|
|
$\partial F_{i, i+1}^{\flat} \subseteq V(K_b) \cup V(K_c)$. Hence
|
|
$F_{i, i+1}^{\flat}$ is a deciding face.
|
|
\end{proof}
|
|
|
|
\begin{remark}[$n_i = 6$ is not handled by the flank face alone]
|
|
\label{rem:n-i-6-flank-fails}
|
|
The natural extension to $n_i = 6$ via the flank face fails ---
|
|
Lemma~\ref{lem:flank-covering-hex} is empirically false in full
|
|
generality. So the flank face $F_{i, i+1}^{\flat}$ does \emph{not}
|
|
yield a structural proof in the $n_i = 6$ case for every
|
|
chord-apex+Kempe colouring, and that case must be handled by other
|
|
faces of the reduced dual.
|
|
\end{remark}
|
|
|
|
We extend the partial proof to handle configurations where neither
|
|
flank face qualifies (i.e., both $n_i, n_{i+1} \ge 7$). The candidate
|
|
is the \emph{outer face} inside $F$, on the merged side of $v_n$.
|
|
|
|
\begin{definition}[Outer face]
|
|
\label{def:outer-face}
|
|
The \emph{outer face} $F_{\mathrm{outer}}^{\flat}$ of
|
|
$\widehat{G}'_{v,i}$ is the face whose boundary is the closed walk
|
|
\[
|
|
v_n \xrightarrow{\text{side-}1} A_{i+2}
|
|
\xrightarrow{\partial F\text{-arc}} A_{i+3}
|
|
\xrightarrow{\text{merged}} A_{i+4}
|
|
\xrightarrow{\partial F\text{-arc}} A_i
|
|
\xrightarrow{\text{side-}0} v_n,
|
|
\]
|
|
i.e., the face inside $F$ bounded by side-$1$, the
|
|
$A_{i+2} \to A_{i+3}$ arc of $\partial F$, the merged edge, the
|
|
$A_{i+4} \to A_i$ arc of $\partial F$, and side-$0$ reversed.
|
|
\end{definition}
|
|
|
|
\begin{lemma}[Outer-face length formula]
|
|
\label{lem:outer-face-length}
|
|
$|F_{\mathrm{outer}}^{\flat}| = n_{i+2} + n_{i+4} - 3$.
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
The boundary length is $1$ (side-$1$) $+\, (n_{i+2} - 3)$ (arc
|
|
$A_{i+2} \to A_{i+3}$) $+\, 1$ (merged) $+\, (n_{i+4} - 3)$ (arc
|
|
$A_{i+4} \to A_i$) $+\, 1$ (side-$0$) $= n_{i+2} + n_{i+4} - 3$.
|
|
\end{proof}
|
|
|
|
\begin{lemma}[Outer-face covering, pentagonal-flanks case]
|
|
\label{lem:outer-face-covering-base}
|
|
If $n_{i+2} = n_{i+4} = 5$ then
|
|
$\partial F_{\mathrm{outer}}^{\flat} \subseteq V(K_b) \cup V(K_c)$.
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
The boundary visits $v_n$, $A_{i+2}$, a single intermediate $P_{23}$
|
|
on the $A_{i+2} \to A_{i+3}$ arc (since $n_{i+2} = 5$ gives arc length
|
|
$2$), $A_{i+3}$, $A_{i+4}$, a single intermediate $P_{40}$ on the
|
|
$A_{i+4} \to A_i$ arc, and $A_i$. The five named vertices are in
|
|
$V(K_b) \cup V(K_c)$ by Lemma~\ref{lem:kempe-spike}; we need only place
|
|
$P_{23}$ and $P_{40}$.
|
|
|
|
By Lemma~\ref{lem:kempe-spike}, $A_{i+3}, A_{i+4} \in V(K_b) \cap V(K_c)$
|
|
via the merged edge. Apply the argument of
|
|
Lemma~\ref{lem:flank-covering-base} at $A_{i+3}$: $A_{i+3}$ has three
|
|
incident edges, one being the merged edge (colour $c$) and the other
|
|
two with colours $\{c_0, c_1\}$. $K_b$ uses merged together with
|
|
$A_{i+3}$'s colour-$c_0$ edge, and $K_c$ uses merged together with its
|
|
colour-$c_1$ edge. The two non-merged edges at $A_{i+3}$ are
|
|
$A_{i+3} P_{23}$ and $A_{i+3} P'$ (where $P'$ lies on the merged-side
|
|
face $F_{\mathrm{merged}}^{\flat}$). Whichever colour
|
|
$\varphi(A_{i+3} P_{23})$ takes, it is in $\{c_0, c_1\}$, so the
|
|
corresponding Kempe cycle walks $A_{i+3} \to P_{23}$, placing
|
|
$P_{23}$ in its vertex set.
|
|
|
|
Symmetrically at $A_{i+4}$ (which is also in $V(K_b) \cap V(K_c)$ via
|
|
merged), the same argument gives $P_{40} \in V(K_b) \cup V(K_c)$.
|
|
\end{proof}
|
|
|
|
\begin{theorem}[Extended partial proof of
|
|
Conjecture~\ref{conj:deciding-face}]
|
|
\label{thm:deciding-face-partial-extended}
|
|
The deciding face exists for $\widehat{G}'_{v,i}$ whenever at least
|
|
one of the following holds:
|
|
\textbf{(a$'$)} $n_i = 5$;
|
|
\textbf{(b$'$)} $n_{i+1} = 5$; or
|
|
\textbf{(c)} $n_{i+2} = n_{i+4} = 5$.
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
Cases (a$'$) and (b$'$) are Theorem~\ref{thm:deciding-face-partial},
|
|
yielding the flank face $F_{i, i+1}^{\flat}$ or $F_{i+1, i+2}^{\flat}$
|
|
as the deciding face. For case (c),
|
|
$|F_{\mathrm{outer}}^{\flat}| = 5 + 5 - 3 = 7 \equiv 1 \pmod 3$ by
|
|
Lemma~\ref{lem:outer-face-length}, and
|
|
$\partial F_{\mathrm{outer}}^{\flat} \subseteq V(K_b) \cup V(K_c)$ by
|
|
Lemma~\ref{lem:outer-face-covering-base}. Hence
|
|
$F_{\mathrm{outer}}^{\flat}$ is a deciding face.
|
|
\end{proof}
|
|
|
|
\begin{remark}[Empirical coverage of
|
|
Theorem~\ref{thm:deciding-face-partial-extended}]
|
|
\label{rem:deciding-face-remaining-case}
|
|
Across all $1{,}586$ $(G, v)$ pairs underlying chord-apex+Kempe
|
|
colourings for $|V(G)| \le 20$, every cyclic neighbour-degree
|
|
sequence around $v$ contains at least one degree-$5$ entry (see
|
|
\texttt{experiments/check\_v\_neighbour\_degrees.py}). Of the
|
|
$7{,}930$ $(G, v, i)$ triples,
|
|
Theorem~\ref{thm:deciding-face-partial-extended} (cases (a$'$),
|
|
(b$'$), (c)) gives a deciding face for $7{,}531$ ($94.97\%$); see
|
|
\texttt{experiments/audit\_tight\_coverage.py}.
|
|
|
|
The remaining $399$ triples ($5.03\%$) have $n_i \ne 5$ \emph{and}
|
|
$n_{i+1} \ne 5$ \emph{and} $(n_{i+2}, n_{i+4}) \ne (5, 5)$, but at
|
|
least one of $\{n_i, n_{i+1}\}$ equals $6$. The flank face on the
|
|
$n_k = 6$ side is the natural candidate (length $5 \not\equiv 0
|
|
\pmod 3$), but Lemma~\ref{lem:flank-covering-hex} is empirically
|
|
false in full generality (boundary coverage fails on $0.92\%$ of
|
|
colourings via Case (b) sub-case (ii)).
|
|
|
|
\textbf{For those colourings the deciding face exists ---
|
|
empirically it is an \emph{original $G'$-face} (a face of $\widehat{G}'_{v,i}$
|
|
inherited from $G'$ and not subdivided by the chord-apex
|
|
construction).} Concretely
|
|
(\texttt{experiments/check\_bad\_subcase\_deciding\_face.py}), every
|
|
single one of the $1{,}314$ chord-apex+Kempe colourings on which
|
|
Lemma~\ref{lem:flank-covering-hex}'s conclusion empirically fails
|
|
has at least one $G'$-pentagon $f$ (length $5 \not\equiv 0 \pmod 3$)
|
|
with $\partial f \subseteq V(K_b) \cup V(K_c)$, making $f$ a
|
|
deciding face. (Independently, $94.06\%$ of the same colourings
|
|
also have $F_{\mathrm{outer}}^{\flat}$ as a deciding face,
|
|
$90.41\%$ have $F_{i+1, i+2}^{\flat}$ as one, and $39.27\%$ have
|
|
$F_{\mathrm{merged}}^{\flat}$ as one --- redundancy is the norm.)
|
|
|
|
This suggests the following structural claim, currently open:
|
|
|
|
\begin{conjecture}[$G'$-pentagon fallback]
|
|
\label{conj:gprime-pentagon-fallback}
|
|
For every chord-apex+Kempe colouring of every reduced dual
|
|
$\widehat{G}'_{v,i}$, at least one $G'$-pentagon $f$ (= pentagonal
|
|
face of $G'$ not equal to $F_v$ or to any of $F_i, \ldots, F_{i+4}$)
|
|
satisfies $\partial f \subseteq V(K_b) \cup V(K_c)$.
|
|
\end{conjecture}
|
|
|
|
\begin{lemma}[Partial proof of
|
|
Conjecture~\ref{conj:gprime-pentagon-fallback} for small uncovered
|
|
set]
|
|
\label{lem:gprime-pigeonhole}
|
|
If $|V(\widehat{G}'_{v,i}) \setminus (V(K_b) \cup V(K_c))| \le 1$,
|
|
then Conjecture~\ref{conj:gprime-pentagon-fallback} holds.
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
The reduced dual contains at least $12 - 1 - 5 = 6$ $G'$-pentagons
|
|
(from the classical lower bound of $12$ pentagons on any cubic plane
|
|
graph with all faces of length $\ge 5$, minus $F_v$ and the five
|
|
faces $F_i, \ldots, F_{i+4}$ that the reduction subdivides). In a
|
|
cubic plane graph each vertex lies on exactly $3$ faces, so a single
|
|
uncovered vertex hits at most $3$ $G'$-pentagons. With at most one
|
|
uncovered vertex, at least $6 - 3 = 3$ $G'$-pentagons have their
|
|
entire boundary in $V(K_b) \cup V(K_c)$.
|
|
\end{proof}
|
|
|
|
\begin{remark}[The pigeonhole stops at $|S| = 1$;
|
|
\textbf{the proof begins to resemble discharging}]
|
|
\label{rem:gprime-pigeonhole-stop}
|
|
Lemma~\ref{lem:gprime-pigeonhole} adds significant structural
|
|
coverage. By Conjecture~\ref{conj:gprime-pentagon-fallback}'s
|
|
empirical check (\texttt{experiments/check\_kb\_kc\_coverage.py}),
|
|
$73.87\%$ of chord-apex+Kempe colourings have $|S| = 0$ and a further
|
|
$\approx 17\%$ have $|S| = 1$, so
|
|
Theorem~\ref{thm:deciding-face-partial-extended} +
|
|
Lemma~\ref{lem:gprime-pigeonhole} cover roughly $91\%$ of
|
|
configurations structurally. The remaining $\approx 9\%$ have
|
|
$|S| \ge 2$, where the trivial pigeonhole no longer forces an
|
|
uncovered $G'$-pentagon to exist (two uncovered vertices can in
|
|
principle hit up to $6$ pentagons). Sharpening
|
|
Lemma~\ref{lem:gprime-pigeonhole} for $|S| \ge 2$ would require finer
|
|
graph-structural input --- specifically, a Kempe-cycle structural
|
|
result about \emph{which} pentagons the uncovered vertices of
|
|
chord-apex+Kempe colourings can occupy. This is exactly the discharging
|
|
flavour of the traditional reducible-configurations approach to the
|
|
Four Colour Theorem (Appel--Haken~\cite{AH77a, AHK77},
|
|
Robertson--Sanders--Seymour--Thomas~\cite{RSST97},
|
|
Gonthier~\cite{Gonthier08}), and we expect the structural case
|
|
analysis to fragment further as $|S|$ increases. We therefore stop
|
|
the case-by-case route here: the residual $\approx 9\%$ structural
|
|
gap is open, and closing it case-by-case is conceivable but no longer
|
|
fundamentally distinct from the classical approach.
|
|
\end{remark}
|
|
|
|
If Conjecture~\ref{conj:gprime-pentagon-fallback} is true, then together
|
|
with Theorem~\ref{thm:deciding-face-partial-extended} it gives a
|
|
structural proof of the deciding-face conjecture for every
|
|
chord-apex+Kempe configuration, hence (via
|
|
Theorem~\ref{thm:deciding-face-implies-conj-5-1}) a structural proof
|
|
of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
in full generality. Empirically this fallback closes the residual
|
|
$0.92\%$ ($1{,}314$ of $142{,}812$) of chord-apex+Kempe colourings
|
|
that the near-spike argument leaves open.
|
|
|
|
Combined, Theorems~\ref{thm:deciding-face-implies-conj-5-1}
|
|
and~\ref{thm:deciding-face-partial-extended} yield a structural proof
|
|
of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
on $7{,}531 / 7{,}930$ ($94.97\%$) of chord-apex+Kempe configurations
|
|
up to $|V(G)| \le 20$; the remaining $5.03\%$ are closed empirically
|
|
via the $G'$-pentagon fallback, but its structural proof
|
|
(Conjecture~\ref{conj:gprime-pentagon-fallback}) is open. The
|
|
deciding-face conjecture itself remains empirically true on all
|
|
$142{,}812$ colourings.
|
|
|
|
The structurally open remaining case is configurations where
|
|
\emph{both} $n_i, n_{i+1} \ge 7$ \emph{and} the merged-side
|
|
degrees $(n_{i+2}, n_{i+4})$ are not both~$5$. We have not observed
|
|
this case empirically for $|V(G)| \le 20$, but it could arise for
|
|
larger triangulations and would then need a further covering lemma
|
|
(extending Lemma~\ref{lem:outer-face-covering-base} to
|
|
$(n_{i+2}, n_{i+4}) \in \{5, 6\}^2$, or a separate argument via
|
|
$F^{\flat}_{\mathrm{merged}}$ of length $n_{i+3} - 2$).
|
|
\end{remark}
|
|
|
|
\begin{remark}[Empirical verification of
|
|
Conjecture~\ref{conj:deciding-face}]
|
|
\label{rem:deciding-face-empirical}
|
|
We have verified
|
|
Conjecture~\ref{conj:deciding-face} computationally on every
|
|
chord-apex+Kempe colouring of every reduced dual of every
|
|
triangulation $G$ of minimum degree $5$ with $|V(G)| \le 20$. Across
|
|
$142{,}812$ such colourings, every single one admits a deciding face.
|
|
The empirical face-length distribution is:
|
|
\begin{center}
|
|
\small
|
|
\begin{tabular}{r|r|r}
|
|
$|f|$ & $|f| \bmod 3$ & \#colourings \\
|
|
\hline
|
|
$4$ & $1$ & $13{,}074$ \\
|
|
$5$ & $2$ & $102{,}498$ \\
|
|
$7$ & $1$ & $18{,}570$ \\
|
|
$8$ & $2$ & $7{,}752$ \\
|
|
$10$ & $1$ & $846$ \\
|
|
$11$ & $2$ & $72$ \\
|
|
\end{tabular}
|
|
\end{center}
|
|
(Every length appearing is $\not\equiv 0 \pmod 3$, as required.)
|
|
See \texttt{experiments/check\_deciding\_face.py}.
|
|
|
|
A natural candidate for the deciding face is one of the two
|
|
``flank'' faces of the reduced dual --- the face bounded by the
|
|
side-$0$ edge, the $A_i \to A_{i+1}$ arc of $\partial F$, and the
|
|
spike edge, or its $A_{i+1} \to A_{i+2}$ counterpart. When the
|
|
parent triangulation's vertex $v$ has a degree-$5$ neighbour $B_k$
|
|
(i.e.\ when the corresponding $G'$-face adjacent to $F_v$ is
|
|
pentagonal), the flank face on that side has length $5 - 1 = 4$ and
|
|
qualifies as a deciding face the moment $V(K_b) \cup V(K_c)$ contains
|
|
its single non-named boundary vertex.
|
|
\end{remark}
|
|
|
|
\begin{remark}[Empirical near-proof of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} via Corollary~\ref{cor:single-cycle-non-constancy}]
|
|
\label{rem:heawood-empirical}
|
|
\sloppy
|
|
By Corollary~\ref{cor:single-cycle-non-constancy},
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
follows from the (a~priori weaker) structural claim:
|
|
\emph{for every chord-apex+Kempe colouring $\varphi$ of every reduced
|
|
dual $\widehat{G}'_{v,i}$, $h_\varphi$ is not constant on $V(K_b)$
|
|
(equivalently, not constant on $V(K_c)$).} We have verified this claim
|
|
computationally on all chord-apex+Kempe colourings of reduced duals
|
|
with $|V(G)| \le 20$ (including the six Holton--McKay duals at
|
|
$n = 21$ as a special case); see
|
|
\texttt{experiments/check\_heawood\_on\_kempe.py} and
|
|
\texttt{experiments/check\_constancy\_obstruction.py}.
|
|
\begin{center}
|
|
\small
|
|
\renewcommand{\arraystretch}{1.15}
|
|
\begin{tabular}{r|r|r|r|l}
|
|
$n$ & \#col.\ tested
|
|
& \#non-const. on $V(K_b)$
|
|
& \#non-const. on $V(K_c)$ & status \\
|
|
\hline
|
|
$14$ & $216$ & $216$ & $216$ & all non-constant \\
|
|
$16$ & $864$ & $864$ & $864$ & all non-constant \\
|
|
$17$ & $4{,}650$ & $4{,}650$ & $4{,}650$ & all non-constant \\
|
|
$18$ & $8{,}070$ & $8{,}070$ & $8{,}070$ & all non-constant \\
|
|
$19$ & $21{,}138$ & $21{,}138$ & $21{,}138$ & all non-constant \\
|
|
$20$ & $107{,}874$ & $107{,}874$ & $107{,}874$ & all non-constant \\
|
|
\hline
|
|
total ($n \le 20$) & $142{,}812$ & $142{,}812$ & $142{,}812$ & \\
|
|
\end{tabular}
|
|
\end{center}
|
|
\noindent In particular, $h_\varphi$ is non-constant on $V(K_b)$ alone
|
|
in every tested colouring (and likewise on $V(K_c)$); by
|
|
Corollary~\ref{cor:single-cycle-non-constancy} each such colouring
|
|
admits a Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
witness. This gives an empirical near-proof of the conjecture for
|
|
$|V(G)| \le 20$ independent of (and consistent with) the direct
|
|
witness-search check of Remark~\ref{rem:conj-3-6-empirical}. A
|
|
structural proof of non-constancy on $V(K_b)$ (or on $V(K_c)$) would
|
|
convert this into a proof of
|
|
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
|
proper.
|
|
\end{remark}
|
|
|
|
\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}
|