From 646cf9d12f9d9d8334b965eced07c4b1d3795d95 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 17 Jun 2026 01:03:37 -0400 Subject: [PATCH] Add connected tire clusters with two-cluster-per-vertex proposition 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 --- .../paper.aux | 35 +++--- .../paper.log | 58 +++++----- .../paper.pdf | Bin 210205 -> 241965 bytes .../paper.tex | 101 ++++++++++++++++++ 4 files changed, 154 insertions(+), 40 deletions(-) diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux index 43d4d2b..1122386 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.aux @@ -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} diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log index a8bd287..e374fe4 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 00:42 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 17 JUN 2026 01:03 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,35 +192,39 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -[2] [3] (./paper.aux) ) +[2] [3] [4] (./paper.aux) ) Here is how much of TeX's memory you used: - 3002 strings out of 478268 - 41895 string characters out of 5846347 - 340223 words of memory out of 5000000 - 21051 multiletter control sequences out of 15000+600000 - 475975 words of font info for 54 fonts, out of 8000000 for 9000 + 3012 strings out of 478268 + 42080 string characters out of 5846347 + 342271 words of memory out of 5000000 + 21059 multiletter control sequences out of 15000+600000 + 476880 words of font info for 57 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,7n,76p,242b,278s stack positions out of 10000i,1000n,20000p,200000b,200000s - - -Output written on paper.pdf (3 pages, 210205 bytes). + 69i,7n,76p,242b,264s stack positions out of 10000i,1000n,20000p,200000b,200000s + + +Output written on paper.pdf (4 pages, 241965 bytes). 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