From 192ad33bd284e40dd2e5dd4441b1dcc940d45726 Mon Sep 17 00:00:00 2001 From: didericis Date: Sat, 23 May 2026 02:41:46 -0400 Subject: [PATCH] dual_decomposition: Kempe-cycle lemma through the spike - Update def:edge-names to distinguish side-0 ({A_i, v_n}) and side-1 ({A_{i+2}, v_n}); merged and spike unchanged. - Add a paragraph defining the {a,b}-Kempe cycle in a 3-edge-coloured cubic graph. - Add lem:kempe-spike: in any proper 3-edge-colouring of the reduced dual, the {c, c_0}-Kempe cycle through the spike contains side-0 and merged (symmetrically for side-1 with c_1). - Proof by Kempe swap: a hypothetical alternative cycle K containing merged but not spike would, after swapping c <-> c_0 on K, give a proper 3-edge-colouring under which spike and merged disagree --- contradicting lem:chord-apex. Co-Authored-By: Claude Opus 4.7 --- .../paper.aux | 3 +- .../paper.log | 52 ++++++++--------- .../paper.pdf | Bin 750077 -> 755306 bytes .../paper.tex | 53 ++++++++++++++++-- 4 files changed, 76 insertions(+), 32 deletions(-) diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.aux b/papers/dual_decomposition_minimal_counterexamples/paper.aux index 7fd54d6..2c8b4dc 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.aux +++ b/papers/dual_decomposition_minimal_counterexamples/paper.aux @@ -12,9 +12,10 @@ \newlabel{lem:chord-apex}{{2.6}{4}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 2.6\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. 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zA%AI2l5yhiN)Dwa$CKf@o!+W#Ub!ah&XYcWKhwu7UA0U)3J_T_)7CPCP4Oz=dg1s> zYcH)|24`pdZ`NU)1!&Q_=XFT!-TTy3X-kw6@6OgwjTqu%%2=e!jP?K<*=+3Tv&HIF z%T`!h$@+@}kbGcadQP%N zDm-Puh|tx8R(QY0!3J7@(IN|T%iJiDjqDGqRDx=3ql8=g1MyhWT?h5|47~JBz}eT1 z5uP>1?Curxe6-58ie#k>=!FTq3CK3|>VM##UbGx-LV-k{CgFms0=|mw&0hJ5`kfHGk@d^r| L($UGP$)WxSwNJ4& diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.tex b/papers/dual_decomposition_minimal_counterexamples/paper.tex index 627a8c3..fc39518 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.tex +++ b/papers/dual_decomposition_minimal_counterexamples/paper.tex @@ -177,11 +177,12 @@ $\widehat{G}'_{v,0}$.} \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}; and the edges $A_iv_n$ and -$A_{i+2}v_n$ are the \emph{side edges}. In the $i = 0$ case of -Figure~\ref{fig:reduced-dual-steps} these are $\{A_3, A_4\}$, $\{A_1, v_n\}$, -and $\{A_0, v_n\}, \{A_2, v_n\}$ respectively. +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 @@ -348,4 +349,46 @@ $G'$ 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} + \end{document}