didericis
b6f2ee8d6e
n=19 (21,138 col., 68s) and n=20 (107,874 col., 361s) both pass. New total for clauses (1)-(4) over n<=20: 142,812/142,812. Also bump max_n in check_conj_3_8_scaled.py default to 20 (was 18) and time_budget_per_n to 7200s. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
36 lines
3.8 KiB
TeX
36 lines
3.8 KiB
TeX
\relax
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{The minimal counterexample}}{1}{}\protected@file@percent }
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\newlabel{lem:triangulate}{{1.1}{1}}
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\newlabel{def:minimal}{{1.2}{1}}
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\newlabel{lem:mindeg}{{1.4}{1}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The reduced dual}}{2}{}\protected@file@percent }
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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{fig:chord-apex-proof}{{2}{5}}
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\newlabel{lem:kempe-spike}{{2.7}{6}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Cubic-graph edge contraction}}{7}{}\protected@file@percent }
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\newlabel{def:cubic-edge-contraction}{{3.1}{7}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Cubic-graph edge contraction (Definition\nonbreakingspace 3.1\hbox {}). Left: a fragment of a cubic plane graph with the contracted edge $e = uv$ highlighted in red. Middle: deleting $e$ leaves $u$ and $v$ of degree\nonbreakingspace $2$. Right: smoothing $u$ and $v$ replaces each pair of incident edges by a single new edge, removing $u, v$ and giving a cubic plane graph again.}}{7}{}\protected@file@percent }
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\newlabel{fig:cubic-edge-contraction}{{3}{7}}
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\newlabel{thm:cubic-contraction-4face}{{3.2}{7}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{8}{}\protected@file@percent }
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\newlabel{sec:toward-4ct}{{4}{8}}
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\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{4.1}{8}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 3.2\hbox {}. Left: the $4$-face $f$ of $H$ under $\varphi $, with the forced colours $\varphi (e_0) = a$, $\varphi (e_1) = b$, $\varphi (e_2) = \varphi (e_3) = c$, $\varphi (w_0) = \varphi (w_1) = b$, and $\varphi (w_2) = \varphi (w_3) = a$. Right: the contracted graph $H'$ under $\varphi '$. The smoothed-in edges $e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is recoloured from $b$ to $c$; every edge outside the face neighbourhood keeps its $\varphi $-colour (dotted in red: the five edges of $H$ removed by the contraction).}}{9}{}\protected@file@percent }
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\newlabel{fig:thm-cubic-contraction-4face}{{4}{9}}
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\newlabel{rem:conj-3-6-empirical}{{4.2}{9}}
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\newlabel{conj:face-monochromatic-pair-strengthened}{{4.3}{10}}
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\newlabel{rem:conj-3-8-empirical}{{4.4}{10}}
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\newlabel{tocindent3}{0pt}
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\newlabel{rem:implication-4ct}{{4.5}{11}}
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\gdef \@abspage@last{11}
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