didericis
0303225f39
- Name the edges of the reduced-dual construction (merged, spike, sides)
via a new definition; use these names in lem:chord-apex.
- Add lem:pentagonal-externals with full exhaustive proof: any proper
3-edge-colouring near a pentagonal face of a cubic plane graph has its
five external edges forming, up to cyclic rotation, the pattern
(a, b, c, c, c) with {a, b, c} = {1, 2, 3} (iff).
- Cite the new lemma in the chord-apex proof scaffold as the lifting step.
- Remove the icosahedron experimental remark.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
19 lines
1.3 KiB
TeX
19 lines
1.3 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{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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\newlabel{lem:chord-apex}{{2.6}{4}}
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\gdef \@abspage@last{4}
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