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dual_decomposition: 4-edge-face criterion, Conj 3.8, cubic contraction theorem

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- Conjecture 3.6: add the 4-edge-face criterion as clause (3), with empirical
  table through n=21 (complete, 535,182/535,182 pass) plus partial n=22
  (641,700 colourings, timed out).
- Conjecture 3.8: strengthening with clause (4) on the b,c-Kempe cycle / 3-colour
  alternative on the new face f_n; existential at the witness level. Tested
  through n=18 (13,800/13,800 pass).
- Definition + figure for cubic-graph edge contraction (delete edge, smooth the
  resulting degree-2 endpoints; equivalent to simple contraction in the dual).
- Theorem: cubic contraction across a 4-face preserves 3-edge-colourability when
  the two opposite boundary edges have different colours. Constructive proof:
  the two smoothed-in edges inherit the colour of the w_i pair they absorb, and
  e_1 is recoloured to the third colour.
- Add 2-panel illustration of the theorem's recolouring.
- Trim Remark 3.7 and 3.9 tables to fit within \textwidth.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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@@ -610,10 +610,249 @@ 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 $\varphi(e_1) = \varphi(e_2)$, and
\item $\varphi(e_1) = \varphi(e_2)$,
\item $e_1$, $e_2$, and the merged edge all lie on a common
$\{a, b\}$-Kempe cycle of $\varphi$.
$\{a, b\}$-Kempe cycle of $\varphi$, and
\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 $e_1$
and $e_2$ by new vertices $X_1, X_2$ and joining them by a new edge
$X_1 X_2$ inside $F$ creates a new face bounded by exactly $4$
edges (the new edge, the two subdivision halves adjacent to it, and
the single $\partial F$-edge between $e_1$ and $e_2$).
\end{enumerate}
\end{conjecture}
\begin{remark}
\label{rem:conj-3-6-empirical}
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 counterexample's
reduced dual is forced to admit such colourings under any proper colouring.
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 tested 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), but
did not finish 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 proper $3$-edge-colouring satisfying chord-apex +
both Kempe-cycle conditions, so the conjecture has no content there. On
every $(G, F, i, \varphi)$ with content, all three clauses of the
conjecture hold simultaneously.
\end{remark}
\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, and the following additional clause holds.
Let $X_1, X_2$ be the new vertices subdividing $e_1, e_2$, joined by a new
edge $X_1 X_2$ inside $F$; write $\widehat{G}'^{+}$ for the resulting
modified graph (which has $|V(\widehat{G}'_{v,i})|+2$ vertices and
$|E(\widehat{G}'_{v,i})|+3$ edges, is again cubic and plane, and admits a
proper $3$-edge-colouring). 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. In particular $\varphi'$ agrees with $\varphi$ on every edge of
$\widehat{G}'_{v,i}$ outside that Kempe cycle, and at $X_1$ and $X_2$ the
two subdivision halves take the colours $\{a, b\}$ in the order forced by
propriety. Write $a := \varphi(e_1) = \varphi(e_2)$,
$c := \varphi'(X_1 X_2)$, and let $b$ be the third colour. Let $f_n$ be
the new $4$-edge face of $\widehat{G}'^{+}$ incident to $X_1 X_2$. Then:
\begin{enumerate}
\setcounter{enumi}{3}
\item 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-3.6-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
$12$ & $1$ & $0$ & --- & vacuous \\
$13$ & $0$ & --- & --- & 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 \\
\hline
total & $23$ & $13{,}800$ & $13{,}800$ & \\
\end{tabular}
\end{center}
\noindent A subtlety: only about half of the Conjecture-3.6-witnesses
individually satisfy clause (4) on each colouring, but in every case some
witness does. The conjecture is therefore an existential statement at the
witness level, not a property of every witness.
\end{remark}
\medskip
The next definition records a cubic-preserving analogue of edge contraction
which turns out --- under planar duality --- to coincide with simple-graph
contraction on the dual side. It will be useful when reasoning about the
modified graph $\widehat{G}'^{+}$ of Conjecture~\ref{conj:face-monochromatic-pair-strengthened}
and its further reductions.
\begin{definition}[Cubic-graph edge contraction]
\label{def:cubic-edge-contraction}
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{cubic-graph edge contraction} of $H$
along $e$ 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{Cubic-graph edge contraction
(Definition~\ref{def:cubic-edge-contraction}). Left: a fragment of a cubic
plane graph with the contracted 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:cubic-edge-contraction}
\end{figure}
\begin{theorem}[Cubic contraction across a 4-face preserves 3-edge-colourability]
\label{thm:cubic-contraction-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
cubic-graph edge contraction of $H$ along $e_0$
(Definition~\ref{def:cubic-edge-contraction}) is well-defined (no loops or
parallel edges are created), then the contracted 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 cubic-graph edge
contraction of $H$ along $e_0$; 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:cubic-contraction-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 contracted 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 contraction).}
\label{fig:thm-cubic-contraction-4face}
\end{figure}
\end{document}