didericis
83e9dba8ac
- Move the iterated-reduction algorithm, its two structural lemmas
(exactly-one-match, all-distinct-exists), and the n=14 trace figure
into a new companion paper at
papers/dual_decomposition_iterated_reduction/. Figures and figure
scripts moved via git mv (history preserved).
- In the main paper, Section 3 ("An iterated reduction") becomes
Section 3 "Cubic-graph edge contraction" (just the contraction
definition + 4-face theorem).
- Restructure Section 4 to host both the original face-monochromatic-pair
conjecture (clauses 1-3) and its strengthening (adds clause 4) as
separate conjectures, after briefly experimenting with folding them
into one. The empirical evidence is asymmetric (n<=21 for (1)-(3),
n<=18 for the full set), which the two-conjecture split presents more
honestly. The companion-paper reference is now in Section 4's intro.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
35 lines
2.4 KiB
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35 lines
2.4 KiB
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Algorithm\nonbreakingspace 2.1\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 and Kempe-cycle conditions of\nonbreakingspace \cite {parent}, 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.}}{3}{}\protected@file@percent }
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