dual_decomposition: swap algorithm trace to n=14 + final-graph conjecture

- Replace the dodecahedron trace at the end of section 3 with the n=14
  triangulation found by search_kempe_property.py: its H_1 admits a
  proper 3-edge-colouring satisfying both chord-apex and Kempe-cycle
  conditions (Lemmas 2.6, 2.7).
- experiments/draw_iterated_reduction_n14.py: rebuilds fig_alg_step{0,1,2}
  with Tutte barycentric layouts (outer face chosen to keep v_n in the
  interior); also runs the algorithm to completion, checking chord-apex +
  Kempe at each step (step 1 satisfies all; step 2 fails chord-apex;
  step 3 terminates).
- Add Conjecture 3.4: G is a minimal counterexample iff no proper
  3-edge-colouring of the final reduced graph H_{t*} has all (spike_t,
  merged_t) pairs in distinct colours.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/dual_decomposition_minimal_counterexamples/paper.aux

@@ -22,6 +22,7 @@
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
-\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Algorithm\nonbreakingspace 3.1\hbox {} on $G' = $ dodecahedron (dual of the icosahedron). \emph  {Left:} $G'$ (20 vertices, 30 edges), with $F_v$ (the inner pentagon) shaded as the face chosen for the first reduction. \emph  {Centre:} $H_1$ (16 vertices, 24 edges) after step\nonbreakingspace (1) with $i_1 = 0$, $3$-edge-coloured by Sage; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker. \emph  {Right:} $H_2$ (12 vertices, 18 edges) after step\nonbreakingspace (3) with $i_t = 0$; the only safe pentagonal face in $H_1$ was the outer pentagon, whose deletion produces $v_n^{(2)}$ and a second chord, giving eight protected edges. No safe pentagonal face remains, so the algorithm terminates. The generating script is \texttt  {experiments/draw\_iterated\_reduction.py}.}}{8}{}\protected@file@percent }
+\@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{conj:no-all-distinct-coloring}{{3.4}{8}}
 \gdef \@abspage@last{8}