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>
52 lines
4.5 KiB
TeX
52 lines
4.5 KiB
TeX
\relax
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\citation{AH77a}
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\citation{AHK77}
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\citation{RSST97}
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\citation{Gonthier08}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
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\newlabel{sec:intro}{{1}{1}}
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\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Organization.}}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Companion paper.}}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The minimal counterexample}}{2}{}\protected@file@percent }
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\newlabel{sec:minimal}{{2}{2}}
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\newlabel{lem:triangulate}{{2.1}{2}}
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\newlabel{def:minimal}{{2.2}{2}}
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\newlabel{lem:mindeg}{{2.4}{3}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{The reduced dual}}{3}{}\protected@file@percent }
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\newlabel{sec:reduced-dual}{{3}{3}}
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\newlabel{def:reduced-dual}{{3.1}{3}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 3.1\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}$.}}{4}{}\protected@file@percent }
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\newlabel{fig:reduced-dual-steps}{{1}{4}}
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\newlabel{def:edge-names}{{3.3}{4}}
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\newlabel{lem:pentagonal-externals}{{3.4}{5}}
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\newlabel{lem:chord-apex}{{3.6}{6}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 3.6\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 3.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{7}{}\protected@file@percent }
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\newlabel{fig:chord-apex-proof}{{2}{7}}
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\newlabel{lem:kempe-spike}{{3.7}{7}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Edge suppression}}{8}{}\protected@file@percent }
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\newlabel{sec:edge-suppression}{{4}{8}}
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\newlabel{def:edge-suppression}{{4.1}{8}}
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\newlabel{thm:edge-suppression-4face}{{4.2}{8}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Edge suppression (Definition\nonbreakingspace 4.1\hbox {}). Left: a fragment of a cubic plane graph with the suppressed 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.}}{9}{}\protected@file@percent }
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\newlabel{fig:edge-suppression}{{3}{9}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{9}{}\protected@file@percent }
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\newlabel{sec:toward-4ct}{{5}{9}}
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\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{9}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 4.2\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 suppressed 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 suppression).}}{10}{}\protected@file@percent }
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\newlabel{fig:thm-edge-suppression-4face}{{4}{10}}
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\newlabel{rem:conj-3-6-empirical}{{5.2}{10}}
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\newlabel{conj:face-monochromatic-pair-strengthened}{{5.3}{11}}
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\newlabel{rem:conj-3-8-empirical}{{5.4}{11}}
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\bibcite{AH77a}{1}
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\bibcite{AHK77}{2}
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\bibcite{RSST97}{3}
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\bibcite{Gonthier08}{4}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent3}{0pt}
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\newlabel{rem:implication-4ct}{{5.5}{12}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent }
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\gdef \@abspage@last{12}
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