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). PDF statistics: - 94 PDF objects out of 1000 (max. 8388607) - 56 compressed objects within 1 object stream + 112 PDF objects out of 1000 (max. 8388607) + 67 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 1 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf index 0e01271..fed3346 100644 Binary files a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf and b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.pdf differ diff --git a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex index 882182a..21e351b 100644 --- a/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex +++ b/papers/heawood_restrictions_on_nested_tire_graph_duals/paper.tex @@ -98,6 +98,107 @@ along the boundary cycles of a nested tire graph, and to formulate a chain-pigeonhole programme in this Heawood labelling parallel to the medial programme of \cite{bauerfeld-medial-tires}. +\section{Connected tire clusters} +\label{sec:tire-clusters} + +The tire treads at a fixed depth partition the depth-$d$ faces of $G$ +\cite{bauerfeld-nested-tires}, but distinct depth-$d$ tires need not be +vertex-disjoint: a single vertex of $G$ may lie on the source-side +boundary of several depth-$d$ tires at once (this occurs exactly when +the depth-$d$ faces around that vertex are split into more than one arc +by depth-$(d{-}1)$ faces). We organise the depth-$d$ tires by this +sharing. + +\begin{lemma}[Same-depth tires meet only in vertices] +\label{lem:same-depth-vertex-meet} +Let $T \neq T'$ be two distinct tire treads at the same depth $d$ in +$\mathcal{T}(G, S)$, arising from connected components $C', C''$ of the +depth-$d$ dual subgraph $G'_d$. Then $T$ and $T'$ share no edge of $G$; +any intersection $V(T) \cap V(T')$ consists of isolated vertices. +\end{lemma} + +\begin{proof} +An edge $e$ of $G$ shared by two depth-$d$ annular faces $f_1, f_2$ is, +by definition of the inner dual \cite[Definition~1.3]{bauerfeld-nested-tires}, +a dual edge of $G'$ joining $d_{f_1}$ and $d_{f_2}$; since $\delta(d_{f_1}) += \delta(d_{f_2}) = d$, this edge lies in $G'_d$, so $d_{f_1}$ and +$d_{f_2}$ belong to the same component of $G'_d$. Hence no edge of $G$ +is shared by annular faces of two \emph{different} components, and +distinct depth-$d$ tires share no edge. Their intersection is therefore +a set of isolated vertices. +\end{proof} + +\begin{definition}[Connected tire cluster] +\label{def:connected-tire-cluster} +Fix a nested tire decomposition $\mathcal{T}(G, S)$ and a depth $d$. On +the set of depth-$d$ tire treads define the relation +\[ + T \sim T' \quad\Longleftrightarrow\quad V(T) \cap V(T') \neq \varnothing . +\] +A \emph{connected tire cluster} at depth $d$ is the subgraph of $G$ +\[ + \mathsf{K} \;=\; \bigcup_{i} T_i \;\subseteq\; G +\] +obtained as the union (of underlying plane graphs) of the tires in a +single connected component $\{T_i\}$ of the transitive closure of +$\sim$. A cluster consisting of a single tire is \emph{trivial}; the +connected tire clusters at depth $d$ partition the depth-$d$ tires. +\end{definition} + +\begin{remark} +\label{rem:cluster-cut-vertices} +By Lemma~\ref{lem:same-depth-vertex-meet} the constituent tires of a +connected tire cluster are joined only at shared vertices, each of which +is a cut vertex of $\mathsf{K}$; a connected tire cluster is thus a +``cactus of tires'' and is in general \emph{not} itself a tire graph, +since the annulus structure of \cite[Definition~1.5]{bauerfeld-nested-tires} +fails at each such pinch. The shared (cut) vertices are precisely the +vertices that belong to more than one depth-$d$ tire. +\end{remark} + +A single vertex may belong to several tires at one depth --- the +high-degree case where its depth-$d$ faces split into many arcs --- so +the number of \emph{tires} through a vertex is unbounded. Clustering +collapses exactly this multiplicity: all tires through a vertex at a +fixed depth share that vertex, hence lie in one cluster. The cluster +count is therefore controlled. + +\begin{proposition}[A vertex meets at most two clusters] +\label{prop:two-clusters-per-vertex} +Every vertex $v \in V(G)$ belongs to at most two connected tire +clusters, namely at most one at each of the two consecutive depths +$\ell_G(v) - 1$ and $\ell_G(v)$. In particular a source vertex +($\ell_G(v) = 0$) belongs to a single cluster. +\end{proposition} + +\begin{proof} +Write $\ell = \ell_G(v)$. + +\emph{Step 1: every bounded face incident to $v$ has dual depth +$\ell - 1$ or $\ell$.} Let $f$ be a bounded triangular face with +$v \in V(f)$. Then +$\delta_G(d_f) = \min_{u \in V(f)} \ell_G(u) \le \ell_G(v) = \ell$. +The other two vertices of $f$ are adjacent to $v$ in $G$, and the level +function $\ell_G(\cdot) = \mathrm{dist}_G(\cdot, S)$ is $1$-Lipschitz +along edges, so each has level at least $\ell - 1$; hence +$\delta_G(d_f) \ge \ell - 1$. Thus $\delta_G(d_f) \in \{\ell-1, \ell\}$ +(only $\delta_G(d_f) = 0$ when $\ell = 0$), so $v$ bounds faces of, and +therefore belongs to tires of, no depth other than $\ell - 1$ or +$\ell$. + +\emph{Step 2: at each depth, all tires through $v$ lie in one cluster.} +Fix $d \in \{\ell-1, \ell\}$ and let $T, T'$ be depth-$d$ tires with +$v \in V(T) \cap V(T')$. Then $V(T) \cap V(T') \ne \varnothing$, so +$T \sim T'$ in the sense of +Definition~\ref{def:connected-tire-cluster}, and all depth-$d$ tires +containing $v$ lie in a single connected component of $\sim$ --- one +connected tire cluster $\mathsf{K}_d$. + +Combining the two steps, $v$ belongs to at most the clusters +$\mathsf{K}_{\ell-1}$ and $\mathsf{K}_{\ell}$, i.e.\ to at most two +connected tire clusters; when $\ell = 0$ only $\mathsf{K}_0$ occurs. +\end{proof} + \section{Heawood restrictions on the tire dual} \label{sec:heawood-restrictions}