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}
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 \newlabel{tocindent1}{17.77782pt}
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-\gdef \@abspage@last{6}
+\gdef \@abspage@last{7}