didericis
c5f81842c7
Add the two-sided cluster decomposition proposition: a vertex's full Heawood face-sum splits as exactly one child-cluster contribution plus one parent-cluster contribution (the at-most-two-clusters bound makes the pairing binary and complete). Explain why this fails per-tire -- a vertex on many same-depth tires has only a fragment of its face-star in any one tire -- and recast the chain-pigeonhole and 4CT conjectures to nested clusters with a cluster restriction relation. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
47 lines
2.0 KiB
TeX
47 lines
2.0 KiB
TeX
\relax
|
|
\citation{bauerfeld-nested-tires}
|
|
\citation{bauerfeld-nested-tires}
|
|
\citation{bauerfeld-nested-tires}
|
|
\citation{bauerfeld-nested-tires}
|
|
\citation{bauerfeld-nested-tires}
|
|
\citation{bauerfeld-nested-tires}
|
|
\citation{Heawood1898}
|
|
\citation{bauerfeld-medial-tires}
|
|
\citation{bauerfeld-nested-tires}
|
|
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
|
|
\citation{bauerfeld-nested-tires}
|
|
\citation{bauerfeld-nested-tires}
|
|
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Connected tire clusters}}{2}{}\protected@file@percent }
|
|
\newlabel{sec:tire-clusters}{{2}{2}}
|
|
\newlabel{lem:same-depth-vertex-meet}{{2.1}{2}}
|
|
\newlabel{def:connected-tire-cluster}{{2.2}{2}}
|
|
\newlabel{rem:cluster-cut-vertices}{{2.3}{2}}
|
|
\newlabel{prop:two-clusters-per-vertex}{{2.4}{2}}
|
|
\citation{bauerfeld-nested-tires}
|
|
\citation{bauerfeld-nested-tires}
|
|
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Heawood restrictions on the tire dual}}{3}{}\protected@file@percent }
|
|
\newlabel{sec:heawood-restrictions}{{3}{3}}
|
|
\newlabel{def:heawood-labelling}{{3.1}{3}}
|
|
\newlabel{rem:no-interior-constraint}{{3.2}{3}}
|
|
\newlabel{def:boundary-sequences}{{3.3}{3}}
|
|
\newlabel{def:heawood-compatible}{{3.4}{3}}
|
|
\newlabel{rem:compat-is-heawood}{{3.5}{4}}
|
|
\newlabel{eq:heawood-face-sum-dual}{{3.1}{4}}
|
|
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Why the programme runs between nested clusters}}{4}{}\protected@file@percent }
|
|
\newlabel{prop:two-sided-decomposition}{{3.6}{4}}
|
|
\bibcite{Heawood1898}{1}
|
|
\bibcite{bauerfeld-depth}{2}
|
|
\bibcite{bauerfeld-nested-tires}{3}
|
|
\bibcite{bauerfeld-medial-tires}{4}
|
|
\bibcite{bauerfeld-nested-tire-duals}{5}
|
|
\newlabel{tocindent-1}{0pt}
|
|
\newlabel{tocindent0}{14.69437pt}
|
|
\newlabel{tocindent1}{17.77782pt}
|
|
\newlabel{tocindent2}{0pt}
|
|
\newlabel{tocindent3}{0pt}
|
|
\newlabel{rem:why-clusters}{{3.7}{5}}
|
|
\newlabel{conj:heawood-chain-pigeonhole}{{3.8}{5}}
|
|
\newlabel{conj:heawood-route-fct}{{3.9}{5}}
|
|
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent }
|
|
\gdef \@abspage@last{5}
|