diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 4029993..7329362 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -25,6 +25,7 @@ \newlabel{fig:inner-dual-annulus-case}{{4}{9}} \newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{9}} \newlabel{rem:bridge-case-theta}{{1.14}{9}} +\newlabel{thm:tread-tree}{{1.15}{10}} \bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-nested-tire-duals}{2} \newlabel{tocindent-1}{0pt} @@ -32,5 +33,7 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{10}{}\protected@file@percent } -\gdef \@abspage@last{10} +\newlabel{rem:tree-multiple-children}{{1.16}{11}} +\newlabel{rem:tree-coloring-factorisation}{{1.17}{11}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{11}{}\protected@file@percent } +\gdef \@abspage@last{11} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index a7d0e98..f23232a 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/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) 27 MAY 2026 02:24 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 02:40 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -511,43 +511,47 @@ Package pdftex.def Info: fig_tire_example.png used on input line 179. LaTeX Warning: `h' float specifier changed to `ht'. -[7] [8] [9] [10] (./paper.aux) ) +[7] [8] [9] +Underfull \vbox (badness 10000) has occurred while \output is active [] + + [10] +[11] (./paper.aux) ) Here is how much of TeX's memory you used: - 14040 strings out of 478268 - 279072 string characters out of 5846347 - 563777 words of memory out of 5000000 - 31865 multiletter control sequences out of 15000+600000 + 14043 strings out of 478268 + 279149 string characters out of 5846347 + 563813 words of memory out of 5000000 + 31868 multiletter control sequences out of 15000+600000 477909 words of font info for 61 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 84i,12n,89p,1156b,803s stack positions out of 10000i,1000n,20000p,200000b,200000s -< -/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb> -Output written on paper.pdf (10 pages, 594243 bytes). +< +/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb> +Output written on paper.pdf (11 pages, 603044 bytes). PDF statistics: - 165 PDF objects out of 1000 (max. 8388607) - 100 compressed objects within 1 object stream + 169 PDF objects out of 1000 (max. 8388607) + 102 compressed objects within 2 object streams 0 named destinations out of 1000 (max. 500000) 23 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 0a605ad..5882ceb 100644 Binary files a/papers/coloring_nested_tire_graphs/paper.pdf and b/papers/coloring_nested_tire_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index f5cd19c..f8b40f7 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -757,6 +757,122 @@ and so contributes no degree-$2$ branch vertex), hence is outerplanar as predicted. \end{remark} +\begin{theorem}[Tire treads form a rooted tree under face containment] +\label{thm:tread-tree} +Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ +and let $S \subseteq V(G)$ be a single-vertex level source +$\{v_0\}$ lying on the outer face of $\Pi_G$. The collection +$\mathcal{R}(G, S)$ of tire treads +(Theorem~\ref{thm:tread-partition}) carries a canonical rooted +tree structure $\mathcal{T}(G, S)$ defined as follows. + +\begin{itemize} +\item \emph{Root.} The depth-$0$ tire tread $T_0$ --- the unique + tire produced by Lemma~\ref{lem:tire-component} at $d = 0$, + with degenerate outer boundary $B_{\mathrm{out}} = \{v_0\}$ + and inner outerplanar graph $O^{(0)} = G[L_1]$ --- is the + root. +\item \emph{Parent.} For each tire tread $T_c$ at depth $d \ge 1$, + its outer boundary $B_{\mathrm{out}}^{(c)}$ is a cycle in + $L_d$. The \emph{parent} of $T_c$ is the unique tire tread + $T_p$ at depth $d - 1$ whose inner outerplanar graph + $O^{(p)}$ has $B_{\mathrm{out}}^{(c)}$ as the boundary cycle + of one of its bounded faces. Equivalently, $R_c$ lies + inside this bounded face of $O^{(p)}$ (which is itself the + region of the plane cut off by $B_{\mathrm{out}}^{(c)}$ on + the side away from $S$). +\item \emph{Children.} The children of a tire tread $T_p$ are in + bijection with those bounded faces of $O^{(p)}$ whose + interiors contain at least one vertex of $G$ at level + $\ge d + 2$ --- equivalently, with the connected components + of $G'_{d+1}$ whose tires have outer boundary cycle equal to + a bounded face of $O^{(p)}$. +\end{itemize} + +Every tire tread except $T_0$ has exactly one parent; a tire +tread may have zero, one, or several children. +\end{theorem} + +\begin{proof} +\emph{Root is well-defined.} At $d = 0$ with single-vertex source +$S = \{v_0\}$, the dual subgraph $G'_0$ is connected (every face +of $G$ incident to $v_0$ has dual depth $0$, and they form a +single fan around $v_0$). By +Lemma~\ref{lem:tire-component}, the unique component of $G'_0$ +gives the depth-$0$ tire $T_0$ described above. + +\emph{Existence of parent.} Fix a tire tread $T_c$ at depth $d +\ge 1$ arising from a connected component $C'_c$ of $G'_d$. Its +outer boundary $B_{\mathrm{out}}^{(c)} = G[V_{C'_c} \cap L_d]$ is +a simple cycle in $L_d$ (Lemma~\ref{lem:tire-component}; the +source-side boundary of a tire is always a simple cycle, by +Proposition~\ref{prop:no-level-d-pinch}). The faces of $G$ +immediately outside $B_{\mathrm{out}}^{(c)}$ on the side facing $S$ +have depth $d - 1$ (one of their three vertices lies in $L_{d-1}$, +two in $L_d$). Let $C'_p$ be the connected component of +$G'_{d-1}$ containing the dual vertex of any such face. + +\emph{Uniqueness of parent.} $B_{\mathrm{out}}^{(c)}$ is a single +simple cycle in $G$, with a well-defined ``$S$-side'' (the side +of the cycle closer to $v_0$ in $\Pi_G$). The depth-$(d-1)$ +faces lying on this side form a single contiguous arc around +$B_{\mathrm{out}}^{(c)}$ in the dual --- they are all $G'$-adjacent +in sequence (each pair of consecutive arc faces shares an edge in +$B_{\mathrm{out}}^{(c)}$). Hence they all lie in the same +connected component $C'_p$ of $G'_{d-1}$, which is therefore +unique. + +\emph{$B_{\mathrm{out}}^{(c)}$ bounds a face of $O^{(p)}$.} The +parent tire $T_p$ has $V(O^{(p)}) = V_{C'_p} \cap L_d \supseteq +V(B_{\mathrm{out}}^{(c)})$. The cycle $B_{\mathrm{out}}^{(c)}$ is +a subgraph of $O^{(p)}$ that bounds a face of $O^{(p)}$ in the +inherited embedding: the cycle traces around a depth-$\ge d+1$ +region (containing $R_c$ and any descendants of $T_c$), which is +exactly a bounded face of $O^{(p)}$. + +\emph{Children description.} The bounded faces of $O^{(p)}$ are +in bijection with the connected components of $G'_d$ whose +faces lie inside those bounded regions (= one component per +bounded face, by an argument analogous to the existence-and- +uniqueness step above, applied one level deeper). + +\emph{Tree property.} Every non-root $T_c$ has a unique parent at +strictly smaller depth. Iterating the parent map strictly +decreases depth, terminating at $T_0$. No cycles can form +(depth is monotone). Hence $\mathcal{T}(G, S)$ is a rooted tree. +\end{proof} + +\begin{remark} +\label{rem:tree-multiple-children} +A parent tire $T_p$ has multiple children precisely when its +inner outerplanar graph $O^{(p)}$ has multiple bounded faces with +non-trivial interiors (= containing depth-$\ge d+2$ vertices of +$G$). This happens, for instance, when $O^{(p)}$ has chords or +cut-vertices that subdivide its inner region, or when $O^{(p)}$ +has multiple connected components in $G[L_{d+1}] \cap V_{C'_p}$. +By contrast, if $O^{(p)}$ is a simple cycle (the spoke-only case +of Remark~\ref{rem:hamilton-cycle-spoke-only}) with a non-empty +interior, $T_p$ has exactly one child. +\end{remark} + +\begin{remark} +\label{rem:tree-coloring-factorisation} +Combining Theorem~\ref{thm:tread-partition} (treads partition +the bounded faces of $G$) with +Theorem~\ref{thm:tread-tree} (treads form a rooted tree), any +proper coloring problem on $G$'s bounded faces factors through: +\begin{itemize} +\item local coloring problems on each tread (the inner dual of + each tread is outerplanar by + Theorem~\ref{thm:inner-dual-outerplanar}), plus +\item consistency constraints along parent-child interfaces (the + cycle $B_{\mathrm{out}}^{(c)}$ shared between a child and the + face of its parent's $O^{(p)}$). +\end{itemize} +This is the structural setup underlying the chain-pigeonhole +program for tire treads. +\end{remark} + \begin{thebibliography}{9}