didericis
41227c6a0f
- Main paper: dual_decomposition_minimal_counterexamples/ -> face_monochromatic_pairs/. Title is now "Face-Monochromatic Pairs and the Four Colour Theorem". - Companion paper: dual_decomposition_iterated_reduction/ -> iterated_reduction_in_reduced_dual/. Title is now "An Iterated Reduction in the Reduced Dual". Its prose and bibliography cite the parent under the new title. - Update one absolute sys.path reference inside check_conj_face_kempe_n15.py that pointed at the old folder. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
35 lines
2.4 KiB
TeX
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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