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dual_decomposition: move strengthened conjecture into Section 4 + 4CT implication

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- Cut Conjecture 3.8 + Remark 3.9 from Section 3 and move into a new
  Section 4 "The Four Colour Theorem from a strengthened conjecture".
- Add Remark 4.X spelling out the implication: clause (4)(i) forces the
  cyclic colour pattern (c,a,c,b) on the new 4-face f_n, two opposite
  edges of which satisfy the hypothesis of Theorem 3.9 verbatim; case
  (ii) is conjecturally reducible to case (i) via a Kempe swap on the
  {b,c}-cycle through X_1 X_2. Theorem 3.9 then produces the proper
  3-edge-colouring of the contraction, contradicting minimality of G.
- Rewrite the bridge prose into the cubic-contraction definition to
  reference Section 4 forward, rather than the conjecture directly.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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@@ -665,78 +665,11 @@ 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.
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}[Cubic-graph edge contraction]
\label{def:cubic-edge-contraction}
@@ -855,4 +788,114 @@ by the contraction).}
\label{fig:thm-cubic-contraction-4face}
\end{figure}
\section{The Four Colour Theorem from a strengthened conjecture}
\label{sec:toward-4ct}
The next conjecture strengthens
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} by adding
a clause that arranges the new $4$-edge face $f_n$ to satisfy the hypotheses
of Theorem~\ref{thm:cubic-contraction-4face}. The strengthening, if true,
would imply the Four Colour Theorem.
\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}
\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:cubic-contraction-4face}: they
lie on the $4$-face $f_n$, share no endpoint, and have different
$\varphi'$-colours. Theorem~\ref{thm:cubic-contraction-4face} then produces
a proper $3$-edge-colouring of the cubic-graph edge contraction
$\widehat{G}'^{+} \!\setminus\! e$ along 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:cubic-contraction-4face} now produces a proper
$3$-edge-colouring of the cubic-graph edge contraction of
$\widehat{G}'^{+}$ along 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}
\end{document}