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Add connected tire clusters with two-cluster-per-vertex proposition

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Define a connected tire cluster (union of same-depth tires joined by
shared vertices, transitive closure), prove same-depth tires meet only
in vertices, and prove every vertex lies in at most two clusters (one at
each of two consecutive depths) -- the bounded coarsening of the
unbounded per-vertex tire count.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-06-17 01:03:37 -04:00
parent 251c453437
commit 646cf9d12f
4 changed files with 154 additions and 40 deletions
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papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux
+22 -13
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@@ -8,14 +8,23 @@
\citation{Heawood1898}
\citation{bauerfeld-medial-tires}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Heawood restrictions on the tire dual}}{2}{}\protected@file@percent }
\newlabel{sec:heawood-restrictions}{{2}{2}}
\newlabel{def:heawood-labelling}{{2.1}{2}}
\newlabel{rem:no-interior-constraint}{{2.2}{2}}
\newlabel{def:boundary-sequences}{{2.3}{2}}
\newlabel{def:heawood-compatible}{{2.4}{2}}
\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}}
\bibcite{Heawood1898}{1}
\bibcite{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tires}{3}
@@ -26,9 +35,9 @@
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\newlabel{rem:compat-is-heawood}{{2.5}{3}}
\newlabel{eq:heawood-face-sum-dual}{{2.1}{3}}
\newlabel{conj:heawood-chain-pigeonhole}{{2.6}{3}}
\newlabel{conj:heawood-route-fct}{{2.7}{3}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent }
\gdef \@abspage@last{3}
\newlabel{rem:compat-is-heawood}{{3.5}{4}}
\newlabel{eq:heawood-face-sum-dual}{{3.1}{4}}
\newlabel{conj:heawood-chain-pigeonhole}{{3.6}{4}}
\newlabel{conj:heawood-route-fct}{{3.7}{4}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
\gdef \@abspage@last{4}