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

- 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>

papers/dual_decomposition_minimal_counterexamples/paper.aux

@@ -20,11 +20,20 @@
 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Algorithm\nonbreakingspace 3.1\hbox {} on $G'=\mathrm  {dual}(G)$, where $G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and $\varphi _1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies both the chord-apex condition (Lemma\nonbreakingspace 2.6\hbox {}) and the Kempe-cycle condition (Lemma\nonbreakingspace 2.7\hbox {}), found by \texttt  {experiments/search\_kempe\_property.py}. \emph  {Left:} $G'$ ($24$ vertices, $36$ edges) with the chosen pentagonal face shaded. \emph  {Centre:} $H_1$ ($20$ vertices, $30$ edges) after step\nonbreakingspace (1) with $i_1 = 1$, $3$-edge-coloured by $\varphi _1$; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share the colour green. \emph  {Right:} $H_2$ ($16$ vertices, $24$ edges) after step\nonbreakingspace (3) with $i_t = 3$; eight edges are protected, and the algorithm terminates one step later (no remaining safe pentagonal face in $H_2$). The generating script is \texttt  {experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte barycentric embeddings with the outer face picked to keep $v_n^{(1)}, v_n^{(2)}$ in the interior.}}{8}{}\protected@file@percent }
 \newlabel{fig:iterated-reduction-trace}{{3}{8}}
 \newlabel{lem:exactly-one-match}{{3.4}{8}}
+\newlabel{lem:all-distinct-exists}{{3.5}{9}}
+\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{3.6}{9}}
+\newlabel{rem:conj-3-6-empirical}{{3.7}{10}}
+\newlabel{conj:face-monochromatic-pair-strengthened}{{3.8}{10}}
+\newlabel{rem:conj-3-8-empirical}{{3.9}{11}}
+\newlabel{def:cubic-edge-contraction}{{3.10}{11}}
+\newlabel{thm:cubic-contraction-4face}{{3.11}{11}}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{0pt}
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\newlabel{lem:all-distinct-exists}{{3.5}{9}}
-\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{3.6}{9}}
-\gdef \@abspage@last{9}
+\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Cubic-graph edge contraction (Definition\nonbreakingspace 3.10\hbox {}). 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\nonbreakingspace $2$. 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.}}{12}{}\protected@file@percent }
+\newlabel{fig:cubic-edge-contraction}{{4}{12}}
+\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 3.11\hbox {}. 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).}}{13}{}\protected@file@percent }
+\newlabel{fig:thm-cubic-contraction-4face}{{5}{13}}
+\gdef \@abspage@last{13}