didericis
192ad33bd2
- Update def:edge-names to distinguish side-0 ({A_i, v_n}) and side-1
({A_{i+2}, v_n}); merged and spike unchanged.
- Add a paragraph defining the {a,b}-Kempe cycle in a 3-edge-coloured cubic
graph.
- Add lem:kempe-spike: in any proper 3-edge-colouring of the reduced dual,
the {c, c_0}-Kempe cycle through the spike contains side-0 and merged
(symmetrically for side-1 with c_1).
- Proof by Kempe swap: a hypothetical alternative cycle K containing merged
but not spike would, after swapping c <-> c_0 on K, give a proper
3-edge-colouring under which spike and merged disagree --- contradicting
lem:chord-apex.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
22 lines
1.9 KiB
TeX
22 lines
1.9 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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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent0}{0pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\gdef \@abspage@last{7}
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