From 00c669818084c95975f21ced513887bb1ef74377 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 01:47:55 -0400 Subject: [PATCH] coloring_nested_tire_graphs: theorem that inner dual of tire tread is outerplanar MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit NEW Theorem 1.12: For any tire graph T, the inner dual Γ of its tire tread (= subgraph of D(T) induced on interior dual vertices) is outerplanar. The theorem also gives a constructive characterization: Γ admits a planar embedding as a (possibly non-simple) Hamilton walk through every d_f, plus zero or more non-crossing chords. Proof structure (constructive): Case 1 (R is a disk, one boundary degenerate): the polygon triangulation has no interior vertex, so its dual is a tree (p-2 vertices, p-3 diagonals). Trees are outerplanar. Case 2 (R is an annulus, both boundaries non-degenerate): Step 1 - Cyclic ordering: cut R along any spoke e* to convert the annulus into a closed disk. The disk boundary traverses B_out + e* + B_in (reverse) + e*, yielding a cyclic sequence S of annular faces with multiplicities (one per boundary edge, + detours for boundary-free faces). Step 2 - Hamilton walk: consecutive entries of S share an interior annular edge or coincide; the resulting closed walk in Γ visits every d_f (using detours for the rare interior annular triangles with zero boundary edges). Step 3 - Non-crossing chords: remaining interior annular edges become chords. Since the underlying E_ann edges in T are non-crossing in the planar embedding, the chords are non-crossing in Γ. Step 4 - Outerplanar layout: place the |F_ann| vertices on a circle in S-order, draw walk edges as the circle, chords inside. All vertices on outer face → outerplanar. Two remarks following: Remark 1.13: spoke-only case is the classical Hamilton cycle Γ ≅ C_{n+m} with zero chords. Remark 1.14: bridge case (O with a bridge whose 2 incident faces are annular) gives the theta graph Θ(1, b, c) — Hamilton cycle of length n + m_∂ plus a single length-1 chord. The length-1 chord contributes no degree-2 branch vertex to a K_{2,3} subdivision, explaining why this is outerplanar despite being a theta graph. Foundational paper grows from 7 to 8 pages. This theorem unlocks the chain pigeonhole argument over tire treads: each tread's coloring problem is on an outerplanar dual graph, where the structure is locally tractable. Co-Authored-By: Claude Opus 4.7 --- papers/coloring_nested_tire_graphs/paper.aux | 9 +- papers/coloring_nested_tire_graphs/paper.log | 60 ++++---- papers/coloring_nested_tire_graphs/paper.pdf | Bin 539071 -> 550882 bytes papers/coloring_nested_tire_graphs/paper.tex | 140 +++++++++++++++++++ 4 files changed, 177 insertions(+), 32 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index f7d34bd..28e3cfa 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -16,6 +16,9 @@ \newlabel{thm:tread-partition}{{1.9}{6}} \newlabel{rem:tire-component-degenerate}{{1.10}{6}} \newlabel{rem:tire-no-extra-hypotheses}{{1.11}{6}} +\newlabel{thm:inner-dual-outerplanar}{{1.12}{7}} +\citation{bauerfeld-nested-tire-duals} +\citation{bauerfeld-nested-tire-duals} \bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-nested-tire-duals}{2} \newlabel{tocindent-1}{0pt} @@ -23,5 +26,7 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent } -\gdef \@abspage@last{7} +\newlabel{rem:hamilton-cycle-spoke-only}{{1.13}{8}} +\newlabel{rem:bridge-case-theta}{{1.14}{8}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent } +\gdef \@abspage@last{8} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 09b6a1f..7ffb02c 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 01:13 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 01:47 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -205,40 +205,40 @@ File: fig_tire_example.png Graphic file (type png) Package pdftex.def Info: fig_tire_example.png used on input line 177. 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z53^Q{h>Op^qzQ3b)~Xrt^fRj##LKHzt%yIJPumdhEn3|{z`9jCLWQ@wi_r7L>K?+Y ze5($Gjn7s&gzvEGMA%*vgq#=JLFM0sax|wLzgA9o;`Ey!86^~- zQ}R+u>Zek)uB6{6nHMigd8}mT?X@}b2+8Vw-ODw|8Lz-gke9h<2ptSFI5aRh3MC~) GPeuxVB))L~ diff --git a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 9578dc8..45b40f9 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -466,6 +466,146 @@ boundary cycle (the link of $v_0$); the corresponding tire graph has degenerate outer boundary $\{v_0\}$. \end{remark} +\begin{theorem}[Inner dual of a tire tread is outerplanar] +\label{thm:inner-dual-outerplanar} +Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph, +and let $\Gamma$ be the graph on vertex set +$\{d_f : f \in F_{\mathrm{ann}}\}$ with an edge $d_f d_{f'}$ for +each interior annular edge of $T$ (= each edge of $T$ whose two +incident faces both lie in $F_{\mathrm{ann}}$). Then $\Gamma$ is +outerplanar. + +Moreover, $\Gamma$ admits a planar embedding as a (possibly +non-simple) Hamilton walk through every $d_f$, plus zero or more +non-crossing chords. +\end{theorem} + +\begin{proof} +We argue by cases on whether the tire tread $R$ is a disk or an +annulus. + +\medskip +\emph{Case 1: $R$ is a closed disk} (one of $B_{\mathrm{out}}, +B_{\mathrm{in}}$ degenerate, by Definition~\ref{def:tire-graph}). +Then $T \cap R$ triangulates a polygon with no interior vertices: +the polygon's boundary cycles round the unique non-degenerate +boundary plus the degenerate apex, and all vertices of $V(T) +\cap R$ lie on this polygon. The dual graph of such a polygon +triangulation is a tree (a classical fact: a triangulation of a +$p$-gon with no interior vertex has $p - 2$ triangles and $p - 3$ +diagonals, and the diagonals' adjacency graph is a tree). Trees +are outerplanar. + +\medskip +\emph{Case 2: $R$ is an annulus} (both $B_{\mathrm{out}}$ and +$B_{\mathrm{in}}$ non-degenerate). We construct an explicit +outerplanar embedding of $\Gamma$ as a Hamilton walk plus +non-crossing chords. + +\medskip +\noindent\emph{Step 1: Cyclic ordering of $F_{\mathrm{ann}}$.} +The boundary of the annular tread is the disjoint union +$\partial R = B_{\mathrm{out}} \sqcup \overline{B_{\mathrm{in}}}$ +(viewing $B_{\mathrm{in}}$ as a closed walk traced in the +appropriate orientation). Each boundary edge of $R$ is incident to +exactly one annular face: walking around $B_{\mathrm{out}}$ in +cyclic order produces a sequence +$f^{\mathrm{out}}_1, f^{\mathrm{out}}_2, \dots, f^{\mathrm{out}}_n$ +of (not necessarily distinct) annular faces, one per +$B_{\mathrm{out}}$-edge; similarly walking around +$B_{\mathrm{in}}$ produces a sequence +$f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{m_\partial}$ where +$m_\partial$ is the length of the inner-boundary walk. Pick any +spoke $e^\star = u w \in E_{\mathrm{ann}}$ with $u \in +V(B_{\mathrm{out}})$ and $w \in V(B_{\mathrm{in}})$; cut $R$ along +$e^\star$. This converts the annulus into a closed disk +$\tilde R$ whose boundary walks once around $B_{\mathrm{out}}$, +once along $e^\star$, once around $B_{\mathrm{in}}$ in reverse, +and once back along $e^\star$. Concatenating the two boundary +sequences (in the order dictated by this disk traversal) yields a +single cyclic sequence +\[ + \mathcal{S} = (f^{\mathrm{out}}_1, \dots, f^{\mathrm{out}}_n, + f^{\mathrm{in}}_1, \dots, f^{\mathrm{in}}_{m_\partial}) +\] +of annular faces with multiplicities. + +\medskip +\noindent\emph{Step 2: The Hamilton walk.} Consecutive entries of +$\mathcal{S}$ correspond either to the same annular face (when two +adjacent boundary edges meet at a vertex incident to a single +annular face) or to two annular faces sharing an interior edge of +$E_{\mathrm{ann}}$. In the former case the walk stays at one +$\Gamma$-vertex; in the latter it uses one $\Gamma$-edge. The +resulting closed walk in $\Gamma$ visits every face that appears +in $\mathcal{S}$ at least once. + +If every $f \in F_{\mathrm{ann}}$ appears in $\mathcal{S}$ (i.e.\ +every annular face has at least one boundary edge of $R$), the walk +is a Hamilton walk in $\Gamma$, and we are done up to Step 3. Each +annular face with two boundary edges contributes a vertex visited +twice; each with three contributes a vertex visited three times. + +If some $f \in F_{\mathrm{ann}}$ does not appear in $\mathcal{S}$ +(i.e.\ has no boundary edge of $R$), then all three edges of $f$ +are interior annular edges, so $d_f$ has degree $3$ in $\Gamma$. +Such a face is ``trapped'' in the interior of the dual graph and +appears as the endpoint of a chord. Extend the walk by: +whenever it crosses an interior annular edge $e$ shared with a +boundary-free face $f$, detour through $f$ and back. After +finitely many such detours (one per boundary-free face), the walk +becomes a Hamilton walk visiting every $d_f$. + +\medskip +\noindent\emph{Step 3: Non-crossing chords.} The $\Gamma$-edges +not used by the Hamilton walk constructed in Step~2 are the +remaining interior annular edges. Each such edge $e \in +E_{\mathrm{ann}}$ corresponds to a chord between two non-adjacent +positions of $\mathcal{S}$. In the inherited planar embedding of +$\Gamma$ in $R$, these chords are drawn as straight segments +between annular triangle centroids; \emph{they do not cross} because +the underlying $E_{\mathrm{ann}}$ edges they cross are themselves +non-crossing in the planar embedding of $T$. + +\medskip +\noindent\emph{Step 4: Outerplanar embedding.} We now lay out +$\Gamma$ as follows: place the $|F_{\mathrm{ann}}|$ vertices on a +circle in the cyclic order given by $\mathcal{S}$ (treating +multiply-visited faces as single circle vertices). Connect +consecutive vertices on the circle by the Hamilton-walk edges, +which forms the closed walk. Draw the remaining edges as chords +inside the circle. Because the chords were non-crossing in $T$'s +planar embedding, they remain non-crossing here. All vertices lie +on the outer face (the unbounded region outside the circle), +making $\Gamma$ outerplanar. $\square$ +\end{proof} + +\begin{remark} +\label{rem:hamilton-cycle-spoke-only} +In the \emph{spoke-only} case (Definition~\ref{def:tire-graph} with +$O$ $2$-connected and $E_{\mathrm{ann}}$ consisting only of spokes), +every annular face has exactly one boundary edge, every +$d_f$ has $\Gamma$-degree $2$, and the construction of the +Theorem~\ref{thm:inner-dual-outerplanar} proof reduces to the +classical Hamilton cycle $\Gamma \cong C_{n+m}$ with zero chords. +\end{remark} + +\begin{remark} +\label{rem:bridge-case-theta} +When $O$ has a bridge $e_{\mathrm{br}} \in E(O)$ whose two +incident faces are annular triangles, $e_{\mathrm{br}}$ contributes +an interior annular edge in $\Gamma$ rather than two leaves in +$D(T)$ (see Definition~1.7 of \cite{bauerfeld-nested-tire-duals}). +The two bridge-incident annular triangles have $\Gamma$-degree $3$; +the resulting $\Gamma$ has the structure of a Hamilton cycle of +length $n + m_\partial$ plus a single chord (length $1$). This +corresponds to the theta graph $\Theta(1, b, c)$ identified +empirically in \cite{bauerfeld-nested-tire-duals}, which has no +$K_{2,3}$ subdivision (since one of the three paths has length $1$ +and so contributes no degree-$2$ branch vertex), hence is +outerplanar as predicted. +\end{remark} + \begin{thebibliography}{9}

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