didericis
464c524fa1
Paper: - Lemmas 3.4 (exactly one match) and 3.5 (all-distinct exists for 4-colourable G) replace the earlier conjecture; both have proofs. - Add Conjecture 3.6: every proper 3-edge-colouring of a counterexample's reduced dual has a face with two same-colour edges that share a Kempe cycle with the merged edge, neither of them being the merged edge. Experiments (all under experiments/): - search_conj_3_6_counterexample.py: finds n=14 tri#1 i_red=0 where the algorithm's phi_t* sits in a Kempe class with no all-distinct colouring (disproves an earlier formulation). - check_kempe_class.py / check_kempe_class_invariance.py / check_kempe_class_monotone.py: Kempe-class counts on H_1 and H_t* for small triangulations; neither monotonicity direction holds. - check_all_distinct_exists.py: even in the conj-3.6 disproof case, H_t* itself admits all-distinct colourings in the *other* Kempe class. - check_constrained_feasibility.py: literal H_t*-interpretation of C1 + K0 + K1 is empirically unsatisfiable (gap in proof strategy noted). - check_conj_face_kempe.py / check_conj_face_kempe_n15.py: test Conj 3.6 on chord-apex+Kempe colourings of reduced duals at n=12, 14, 15; 216/216 colourings on n=14 satisfy the conjecture, others vacuous. - draw_step1_conj36.py: figure showing a Conj 3.6 witness on H_1 with two new vertices on the witness edges and a new red bridge between them. - draw_step1_conj36_recolored.py: same but with the Kempe cycle recoloured alternately from merged so propriety holds. - draw_lift_to_Gprime.py: lifts the modified+recoloured H_1 back to a proper 3-edge-colouring of the modified G' (24+2 vertices, 39 edges, same Tutte layout as figure 3's first graphic so positions line up). Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
31 lines
3.5 KiB
TeX
31 lines
3.5 KiB
TeX
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\newlabel{lem:triangulate}{{1.1}{1}}
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\newlabel{lem:mindeg}{{1.4}{1}}
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\newlabel{def:reduced-dual}{{2.1}{2}}
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\newlabel{def:edge-names}{{2.3}{2}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 2.1\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{3}{}\protected@file@percent }
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\newlabel{fig:reduced-dual-steps}{{1}{3}}
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\newlabel{lem:pentagonal-externals}{{2.4}{3}}
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\newlabel{lem:chord-apex}{{2.6}{4}}
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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 }
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\newlabel{lem:kempe-spike}{{2.7}{6}}
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\newlabel{alg:iterated-reduction}{{3.1}{7}}
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\newlabel{rem:alg-invariants}{{3.2}{7}}
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\newlabel{rem:alg-chord-apex}{{3.3}{7}}
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\@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 }
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\newlabel{fig:iterated-reduction-trace}{{3}{8}}
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\newlabel{lem:all-distinct-exists}{{3.5}{9}}
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\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{3.6}{9}}
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