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dual_decomposition: iterated-reduction algorithm + Kempe/chord-apex search

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- Add section 3 with Algorithm 3.1 (iterated reduction with protected edges)
  and remarks on invariants and chord-apex applicability.
- Add fig:iterated-reduction-trace illustrating the algorithm on G' =
  dodecahedron (G' -> H_1 -> H_2 -> terminate).
- experiments/iterated_reduction.py: Sage implementation of the algorithm.
- experiments/draw_iterated_reduction.py: produces the 3 trace figures.
- experiments/check_dodecahedron_kempe.py: enumerate proper 3-edge-colorings
  of the dodecahedron's reduced dual and check the chord-apex + Kempe-cycle
  conditions (0 of 36 colorings satisfy all three).
- experiments/search_kempe_property.py: search across min-deg-5
  triangulations; the n = 14 first plantri triangulation is the smallest hit
  (reduced dual has 20 v, 30 e).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-23 12:40:38 -04:00
parent 192ad33bd2
commit c987259c14
11 changed files with 1057 additions and 55 deletions
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papers/dual_decomposition_minimal_counterexamples/paper.aux
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\@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}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{An iterated reduction}}{7}{}\protected@file@percent }
\newlabel{alg:iterated-reduction}{{3.1}{7}}
\newlabel{rem:alg-invariants}{{3.2}{7}}
\newlabel{rem:alg-chord-apex}{{3.3}{7}}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{0pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\gdef \@abspage@last{7}
\@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 }
\newlabel{fig:iterated-reduction-trace}{{3}{8}}
\gdef \@abspage@last{8}