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 <noreply@anthropic.com>

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 $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 2.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{5}{}\protected@file@percent }
 \newlabel{fig:chord-apex-proof}{{2}{5}}
+\newlabel{lem:kempe-spike}{{2.7}{6}}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{0pt}
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\gdef \@abspage@last{6}
+\gdef \@abspage@last{7}

papers/dual_decomposition_minimal_counterexamples/paper.log

@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  22 MAY 2026 20:45
+This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  23 MAY 2026 02:40
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
@@ -226,50 +226,50 @@ uced_dual_step3.png> <./fig_reduced_dual_step4.png>]
 <fig_chord_apex_step1.png, id=34, 505.03976pt x 502.06393pt>
 File: fig_chord_apex_step1.png Graphic file (type png)
 <use fig_chord_apex_step1.png>
-Package pdftex.def Info: fig_chord_apex_step1.png  used on input line 286.
+Package pdftex.def Info: fig_chord_apex_step1.png  used on input line 287.
 (pdftex.def)             Requested size: 251.9989pt x 250.5104pt.
 <fig_chord_apex_step2.png, id=35, 490.16064pt x 499.51323pt>
 File: fig_chord_apex_step2.png Graphic file (type png)
 <use fig_chord_apex_step2.png>
-Package pdftex.def Info: fig_chord_apex_step2.png  used on input line 287.
+Package pdftex.def Info: fig_chord_apex_step2.png  used on input line 288.
 (pdftex.def)             Requested size: 172.79846pt x 176.08986pt.
 <fig_chord_apex_step3.png, id=36, 490.16064pt x 499.51323pt>
 File: fig_chord_apex_step3.png Graphic file (type png)
 <use fig_chord_apex_step3.png>
-Package pdftex.def Info: fig_chord_apex_step3.png  used on input line 288.
+Package pdftex.def Info: fig_chord_apex_step3.png  used on input line 289.
 (pdftex.def)             Requested size: 172.79846pt x 176.08986pt.
 
 
 LaTeX Warning: `h' float specifier changed to `ht'.
 
 [4] [5 <./fig_chord_apex_step1.png> <./fig_chord_apex_step2.png> <./fig_chord_a
-pex_step3.png>] [6] (./paper.aux) ) 
+pex_step3.png>] [6] [7] (./paper.aux) ) 
 Here is how much of TeX's memory you used:
- 3042 strings out of 478268
+ 3043 strings out of 478268
- 43139 string characters out of 5846347
+ 43156 string characters out of 5846347
- 343136 words of memory out of 5000000
+ 343146 words of memory out of 5000000
- 21083 multiletter control sequences out of 15000+600000
+ 21084 multiletter control sequences out of 15000+600000
  476364 words of font info for 55 fonts, out of 8000000 for 9000
  1302 hyphenation exceptions out of 8191
  69i,8n,76p,664b,298s stack positions out of 10000i,1000n,20000p,200000b,200000s
-</usr/local/texlive/2022/texmf-dist/fonts/ty
+</usr/local/texlive/2022/texmf-dist/font
-pe1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
+s/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts
-e1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
+/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts
-e1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type
+/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/
-1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
+type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/t
-/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/p
+ype1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/typ
-ublic/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pub
+e1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1
-lic/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
+/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/pu
-/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
+blic/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/publi
-sfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
+c/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public
-fonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
+/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
-nts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfon
+msfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
-ts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts
+sfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsf
-/symbols/msam10.pfb>
+onts/symbols/msam10.pfb>
-Output written on paper.pdf (6 pages, 750077 bytes).
+Output written on paper.pdf (7 pages, 755306 bytes).
 PDF statistics:
- 103 PDF objects out of 1000 (max. 8388607)
+ 108 PDF objects out of 1000 (max. 8388607)
- 54 compressed objects within 1 object stream
+ 58 compressed objects within 1 object stream
  0 named destinations out of 1000 (max. 500000)
  36 words of extra memory for PDF output out of 10000 (max. 10000000)
 

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
+are named as follows. The chord $A_{i+3}A_{i+4}$ is the \emph{merged edge};
-edge $A_{i+1}v_n$ is the \emph{spike edge}; and the edges $A_iv_n$ and
+the edge $A_{i+1}v_n$ is the \emph{spike edge}; the edge $A_iv_n$ is the
-$A_{i+2}v_n$ are the \emph{side edges}. In the $i = 0$ case of
+\emph{side-$0$ edge}; and the edge $A_{i+2}v_n$ is the \emph{side-$1$ edge}.
-Figure~\ref{fig:reduced-dual-steps} these are $\{A_3, A_4\}$, $\{A_1, v_n\}$,
+In the $i = 0$ case of Figure~\ref{fig:reduced-dual-steps} these are
-and $\{A_0, v_n\}, \{A_2, v_n\}$ respectively.
+$\{A_3, A_4\}$, $\{A_1, v_n\}$, $\{A_0, v_n\}$, 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}