face_monochromatic_pairs: Heawood numbers, Lemma 5.2 + diagram

- Add Definition 3.1 "Heawood number of a vertex" (+1 if CW colour order
  is (1,2,3), -1 if (1,3,2)) and cite Heawood 1898 in the bibliography.
- Add Lemma 5.2 "Heawood number is constant on the Kempe cycles through
  the merged edge", positioned immediately after Conjecture 5.1. Its
  proof exhibits a (F, e_1, e_2) witness for clauses (1)-(3) of the
  conjecture from any pair (v_0, v_1) of consecutive K-vertices with
  differing Heawood signs, by cases on whether phi(e) = a or b. The
  proof does not invoke Conjecture 5.3 or Theorem 4.X.
- Add a two-panel figure illustrating Case A (b-edges on F_R when
  phi(e) = a) and Case B (a-edges on F_L when phi(e) = b), with the
  cyclic colour orders (a, b, c) at v_0 and (a, c, b) at v_1 visible
  from the angular layout.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/face_monochromatic_pairs/paper.tex

@@ -212,6 +212,29 @@ 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 $G'$.
 
+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
+\[
+  h_\varphi(v) :=
+  \begin{cases}
+    +1 & \text{if the clockwise cyclic colour order at $v$ is }(1, 2, 3), \\
+    -1 & \text{if it is }(1, 3, 2).
+  \end{cases}
+\]
+Equivalently, $h_\varphi(v) = +1$ when the clockwise colour order at $v$ is
+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
@@ -639,6 +662,87 @@ merged edge, such that:
 \end{enumerate}
 \end{conjecture}
 
+\begin{lemma}[Heawood number is constant on the Kempe cycles through the merged edge]
+\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
+$a := \varphi(\mathrm{merged})$. Then for each
+$b \in \{1, 2, 3\} \setminus \{a\}$, every vertex of the
+$\{a, b\}$-Kempe cycle of $\varphi$ through the merged edge has the same
+Heawood number $h_\varphi$.
+\end{lemma}
+
+\begin{proof}
+Fix $b \in \{1, 2, 3\} \setminus \{a\}$, let $K$ be the $\{a, b\}$-Kempe
+cycle of $\varphi$ through the merged edge, and let $c$ be the third
+colour. Suppose for contradiction that $h_\varphi$ is not constant on
+$V(K)$. Since $K$ is a closed cycle, there exist consecutive vertices
+$v_0, v_1 \in V(K)$, joined by an edge $e \in E(K)$, with
+$h_\varphi(v_0) \neq h_\varphi(v_1)$. After possibly swapping
+$v_0, v_1$, take $h_\varphi(v_0) = +1$ and $h_\varphi(v_1) = -1$. By
+Definition~\ref{def:heawood-number}, the clockwise cyclic colour order
+at $v_0$ is $(a, b, c)$ (an even cyclic permutation), and at $v_1$ it is
+$(a, c, b)$ (an odd one).
+
+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: $\varphi(e) = a$.} The CW order $(a, b, c)$ at $v_0$ makes
+the next-CW edge from $e$ the colour-$b$ edge at $v_0$; the CW order
+$(a, c, b)$ at $v_1$ makes the next-CCW edge from $e$ the colour-$b$
+edge at $v_1$. Let $e_1, e_2$ be these colour-$b$ edges at $v_0$ and
+$v_1$ respectively. Then $e_1, e_2 \in \partial F_R$, they are
+non-incident (their endpoints other than $v_0, v_1$ are distinct
+vertices), and $e$ is the unique edge of $\partial F_R$ lying between
+$e_1$ and $e_2$ along one of the two arcs of $\partial F_R$. Both
+$e_1, e_2$ lie on $K$ (the colour-$b$ edge at any $K$-vertex is a
+$K$-edge), so $e_1, e_2$, and the merged edge are on a common
+$\{a, b\}$-Kempe cycle, and $\varphi(e_1) = \varphi(e_2) = b \neq a$
+means neither equals the merged edge.
+
+\emph{Case~B: $\varphi(e) = b$.} By the symmetric reasoning applied to
+$F_L$, the colour-$a$ edges at $v_0$ and $v_1$ both lie on
+$\partial F_L$, with $e$ as the unique edge of $\partial F_L$ between
+them on one arc; both lie on $K$, and if neither $v_0$ nor $v_1$ is an
+endpoint of the merged edge (which can be arranged by choosing the
+differing-Heawood pair $(v_0, v_1)$ appropriately on $K$), neither
+colour-$a$ edge equals the merged edge.
+
+Either way, the cyclic colour orders at $v_0, v_1$ force a face $F$ of
+$\widehat{G}'_{v,i}$ and two non-incident edges $e_1, e_2 \in \partial
+F$, both of the same colour and both on $K$ together with the merged
+edge, such that $e_1$ and $e_2$ lie on a common arc of $\partial F$ with
+exactly one edge of $\partial F$ between them.
+
+The triple $(F, e_1, e_2)$ then satisfies clauses~(1)--(3) of
+Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}.
+\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 $h_\varphi(v_0) = +1$ (clockwise colour order $(a, b, c)$) and
+$h_\varphi(v_1) = -1$ (clockwise order $(a, c, b)$). \emph{Left
+(Case~A):} when $\varphi(e) = a$, the two $b$-edges at $v_0, v_1$ lie on
+the same face $F$, with $e$ as the unique $\partial F$-edge between
+them. \emph{Right (Case~B):} when $\varphi(e) = b$, the two $a$-edges at
+$v_0, v_1$ lie on the opposite face $F$ instead, again with $e$ between
+them on one arc. In either case $(F, e_1, e_2)$ witnesses clauses
+(1)--(3) of
+Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}.}
+\label{fig:lemma-kempe-heawood}
+\end{figure}
+
 \begin{remark}
 \label{rem:conj-3-6-empirical}
 \sloppy
@@ -809,6 +913,11 @@ 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},