Add Kempe-balanced colouring definition and validity classifier

Define Kempe-balanced colourings of a full medial tire graph (Def 5.7):
for each valid face (outer face or interior non-tooth face of B(T)) and
each colour pair {a,b}, the number of tooth apexes incident to the face
coloured a or b must be even.  Add Remark 5.8 (necessity: a colouring of
M(T) extends to M(G) only if it is Kempe-balanced) and rename Lemma 5.5
to "Kempe chains are cycles".

Add kempe_valid_colorings.py: enumerate all proper 3-colourings of a full
medial tire graph, label each Kempe-balanced/valid or invalid, and plot
them with the offending face's Kempe chains and odd apex set highlighted
on invalid panels.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/medial_tire_decompositions_of_plane_triangulations/paper.tex

@@ -727,7 +727,8 @@ Since $M$ is $4$-regular and $\varphi$ is proper, every vertex of
 $M_P$ has degree $2$ in $M_P$.  Hence every component of $M_P$ is a
 cycle.  We call these components the $P$-Kempe cycles of $\varphi$.
 
-\begin{lemma}[Kempe cycles are cycles]
+\begin{lemma}[Kempe chains are cycles]
+\label{lem:kempe-cycles}
 Let $G$ be a plane triangulation, let $M=M(G)$, and let
 $\varphi$ be a proper $3$-colouring of $M$.  For each
 $P\in\{\{1,2\},\{2,3\},\{3,1\}\}$, every component of $M_P$ is a cycle.
@@ -763,6 +764,7 @@ in the tire tree.
 The following lemma is the basic conservation principle.
 
 \begin{lemma}[Kempe-cycle conservation across level cycles]
+\label{lem:kempe-conservation}
 Let $C$ be a level cycle of $M$ separating a parent side from a child
 side.  Let $K$ be a $P$-Kempe cycle for some
 $P\in\{\{1,2\},\{2,3\},\{3,1\}\}$.  Then $K$ cannot enter the child side
@@ -784,6 +786,56 @@ $C$.  Thus every entrance through $C$ is paired with an exit through
 $C$.
 \end{proof}
 
+We now use these Kempe cycles to single out the colourings of a full
+medial tire graph that respect the annular tooth structure.
+
+\begin{definition}[Kempe-balanced colouring]
+\label{def:kempe-balanced}
+Let $\varphi$ be a proper $3$-colouring of the full medial tire graph
+$\mathsf{M}(T)$.  For a colour pair $P=\{a,b\}$, let $\mathsf{M}(T)_P$ be
+the subgraph induced by the vertices of colours $a$ and $b$.  Since
+$\mathsf{M}(T)$ need not be $4$-regular, the components of
+$\mathsf{M}(T)_P$ are paths or cycles; we call them the $P$-\emph{Kempe
+chains} of $\varphi$.  Every vertex of colour $a$ or $b$ lies on exactly
+one $P$-Kempe chain.
+
+A \emph{valid face} is the outer face of $\mathsf{M}(T)$, or an interior
+face of $B(T)$ that is not a tooth---namely the root face or a bite
+inner-gap face of Remark~\ref{rem:bite-face-count}.  The \emph{tooth
+apexes incident to} a valid face $F$ are:
+\begin{itemize}
+  \item the up-tooth apexes (Definition~\ref{def:annular-teeth}), when
+  $F$ is the outer face;
+  \item the singleton down-tooth apexes whose annular edge lies on $F$,
+  when $F$ is interior---the apex on annular edge $m$ being incident to
+  the innermost bite $(i,j)$ with $i<m<j$, or to the root face if there
+  is none.
+\end{itemize}
+Bite apexes are never incident to a valid face in this sense.
+
+For a colour pair $P=\{a,b\}$ write $\nu_P(F)$ for the number of tooth
+apexes incident to $F$ that are coloured $a$ or $b$---equivalently, that
+lie on a $P$-Kempe chain.  The colouring $\varphi$ is
+\emph{Kempe-balanced} if $\nu_P(F)$ is even for every valid face $F$ and
+every colour pair $P$.
+\end{definition}
+
+\begin{remark}[Necessity of Kempe-balance]
+\label{rem:kempe-balance-necessary}
+A proper $3$-colouring of $\mathsf{M}(T)$ can be part of a proper
+$3$-colouring of the whole medial graph $M(G)$ only when it is
+Kempe-balanced: if $\varphi$ is the restriction to $\mathsf{M}(T)$ of a
+proper $3$-colouring of $M(G)$, then $\varphi$ is Kempe-balanced.
+Equivalently, a colouring of $\mathsf{M}(T)$ that fails the parity
+condition at some valid face and colour pair cannot extend to a proper
+$3$-colouring of $M(G)$.  This is an instance of Kempe-cycle
+conservation (Lemma~\ref{lem:kempe-conservation}): in the $4$-regular
+graph $M(G)$ each $P$-Kempe chain of $\mathsf{M}(T)$ closes up into a
+$P$-Kempe cycle, and such a cycle crosses the boundary separating a
+valid face from the rest of the sphere an even number of times, so the
+$P$-coloured apexes incident to that face occur in even number.
+\end{remark}
+
 More generally, let $T$ be a medial tire region with boundary
 \[
   \partial T = C_0\sqcup C_1\sqcup\cdots\sqcup C_m.