diff --git a/papers/coloring_nested_tire_dual_graphs/paper.aux b/papers/coloring_nested_tire_dual_graphs/paper.aux index a258f40..4696eed 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.aux +++ b/papers/coloring_nested_tire_dual_graphs/paper.aux @@ -1,41 +1,43 @@ \relax +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } -\newlabel{def:dual-depth}{{1.4}{1}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent } -\newlabel{fig:dual-depth}{{1}{2}} -\newlabel{def:tire-graph}{{1.5}{2}} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent } -\newlabel{fig:tire-example}{{2}{3}} -\newlabel{rem:tire-counts}{{1.6}{3}} -\newlabel{def:partial-tire-dual}{{1.7}{3}} -\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.8\hbox {}.}}{4}{}\protected@file@percent } -\newlabel{fig:partial-tire-dual-example}{{3}{4}} -\newlabel{prop:partial-tire-dual-structure}{{1.8}{4}} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.8\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{5}{}\protected@file@percent } -\newlabel{fig:partial-tire-dual-bridge}{{4}{5}} -\newlabel{prop:no-level-d-pinch}{{1.9}{5}} +\newlabel{prop:partial-tire-dual-structure}{{1.1}{1}} +\citation{bauerfeld-nested-tires} +\newlabel{prop:no-level-d-pinch}{{1.2}{2}} +\citation{bauerfeld-nested-tires} \citation{bauerfeld-depth} -\newlabel{lem:tire-component}{{1.10}{6}} +\citation{bauerfeld-nested-tires} +\newlabel{lem:tire-component}{{1.3}{3}} \citation{bauerfeld-depth} -\newlabel{rem:tire-component-degenerate}{{1.11}{8}} -\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}} -\newlabel{prop:edge-vertex-bijection}{{1.13}{8}} -\newlabel{rem:edge-vertex-corollary}{{1.14}{9}} -\newlabel{def:tire-annular-subgraph}{{1.15}{9}} -\newlabel{def:tire-annular-face-connector}{{1.16}{9}} -\newlabel{def:spokes}{{1.17}{9}} -\newlabel{rem:facial-dual-spoke-only}{{1.18}{9}} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{10}{}\protected@file@percent } -\newlabel{fig:facial-dual-choices}{{5}{10}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{10}{}\protected@file@percent } -\newlabel{sec:latin-conjecture}{{2}{10}} +\citation{bauerfeld-nested-tires} +\newlabel{rem:tire-component-degenerate}{{1.4}{4}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.5}{4}} +\newlabel{prop:edge-vertex-bijection}{{1.6}{4}} +\citation{bauerfeld-nested-tires} +\citation{bauerfeld-nested-tires} +\newlabel{rem:edge-vertex-corollary}{{1.7}{5}} +\newlabel{def:tire-annular-subgraph}{{1.8}{5}} +\newlabel{def:tire-annular-face-connector}{{1.9}{5}} +\newlabel{def:spokes}{{1.10}{6}} +\newlabel{rem:facial-dual-spoke-only}{{1.11}{6}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{6}{}\protected@file@percent } +\newlabel{sec:latin-conjecture}{{2}{6}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{7}{}\protected@file@percent } +\newlabel{fig:facial-dual-choices}{{1}{7}} +\newlabel{conj:latin}{{2.1}{7}} +\newlabel{conj:chain-latin}{{2.2}{7}} \bibcite{bauerfeld-depth}{1} +\bibcite{bauerfeld-nested-tires}{2} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{conj:latin}{{2.1}{11}} -\newlabel{conj:chain-latin}{{2.2}{11}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{11}{}\protected@file@percent } -\gdef \@abspage@last{11} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent } +\gdef \@abspage@last{8} diff --git a/papers/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index 82fbb25..b9e1bdf 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.log +++ b/papers/coloring_nested_tire_dual_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 00:44 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:54 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -191,52 +191,20 @@ Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) - -File: fig_dual_depth.png Graphic file (type png) - -Package pdftex.def Info: fig_dual_depth.png used on input line 107. -(pdftex.def) Requested size: 251.9989pt x 237.67276pt. - - -LaTeX Warning: `h' float specifier changed to `ht'. - [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -[2 <./fig_dual_depth.png>] - -File: fig_tire_example.png Graphic file (type png) - -Package pdftex.def Info: fig_tire_example.png used on input line 161. -(pdftex.def) Requested size: 280.79956pt x 188.56097pt. - [3 <./fig_tire_example.png>] - -File: fig_partial_tire_dual.png Graphic file (type png) - -Package pdftex.def Info: fig_partial_tire_dual.png used on input line 226. -(pdftex.def) Requested size: 280.79956pt x 233.36552pt. - -File: fig_partial_tire_dual_bridge.png Graphic file (type png) - -Package pdftex.def Info: fig_partial_tire_dual_bridge.png used on input line 2 -41. -(pdftex.def) Requested size: 306.0022pt x 204.59406pt. - - -LaTeX Warning: `h' float specifier changed to `ht'. - -[4 <./fig_partial_tire_dual.png>] [5 <./fig_partial_tire_dual_bridge.png>] -[6] [7] [8] - -LaTeX Warning: Reference `def:dual' on page 9 undefined on input line 577. - -[9] - +[2] [3] [4] [5] + File: notes/fig_facial_dual_choices.png Graphic file (type png) Package pdftex.def Info: notes/fig_facial_dual_choices.png used on input line -656. +495. 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PDF statistics: - 136 PDF objects out of 1000 (max. 8388607) - 78 compressed objects within 1 object stream + 144 PDF objects out of 1000 (max. 8388607) + 87 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) - 26 words of extra memory for PDF output out of 10000 (max. 10000000) + 6 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_dual_graphs/paper.pdf b/papers/coloring_nested_tire_dual_graphs/paper.pdf index 5e5bd1f..4c5e064 100644 Binary files a/papers/coloring_nested_tire_dual_graphs/paper.pdf and b/papers/coloring_nested_tire_dual_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_dual_graphs/paper.tex b/papers/coloring_nested_tire_dual_graphs/paper.tex index 043f7dc..2b4ac16 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.tex +++ b/papers/coloring_nested_tire_dual_graphs/paper.tex @@ -44,7 +44,18 @@ \dedicatory{} \begin{abstract} -% TODO: abstract. +This is a follow-up to \cite{bauerfeld-nested-tires}, which +establishes the basic vocabulary of tire graphs $T$ and their +partial tire duals $D(T)$. Building on those definitions, we +analyse the structure of $D(T)$ in the spoke-only case (a corona +graph $C_{n+m} \circ K_1$), prove the tire-component lemma +exhibiting every BFS-level component as a tire graph, give an +edge-vertex coloring bijection that reduces counting proper +$3$-edge-colorings of $D(T)$ to counting proper $3$-vertex-colorings +of a cycle, and develop the tire-annular-subgraph, face-connector, +and inner/outer-spoke structures in $G'$. A concluding section +records a Latin-substructure conjecture for chain-pigeonhole +compatibility of adjacent tires. \end{abstract} \maketitle @@ -58,205 +69,33 @@ minimal counterexample to the Four Colour Theorem -- a smallest triangulation admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph admitting no proper $3$-edge-colouring. -We study the structure such a minimal counterexample would have to exhibit -through the lens of \emph{nested level duals}. Fixing a level source $S$ in $G$ -endows the dual $G'$ with a Breadth-First-Search--derived labelling, the dual -depth of Definition~\ref{def:dual-depth}, and the level structure of $G$ organises -$G'$ into a family of nested cycles carrying these labels. Our aim is to express -the obstruction to a $3$-edge-colouring of $G'$ as conditions on this nested -labelled-cycle structure. +This paper is the second in a series studying that structure +through the lens of \emph{nested level duals}. The foundational +vocabulary --- level sources, levels, the inner planar dual $G'$ +and its dual depth, tire graphs, and partial tire duals +$D(T)$ --- is developed in the companion paper +\cite{bauerfeld-nested-tires}; we refer to that paper for all +basic definitions and rely on them throughout. In particular we +use, without restating, the notions of: +\begin{itemize} + \item \emph{level source} $S$ and $G$-vertex levels $\ell_G(v)$; + \item the inner planar dual $G'$ + (\cite[Definition~1.3]{bauerfeld-nested-tires}); + \item \emph{dual depth} $\delta_G(d_f)$ + (\cite[Definition~1.4]{bauerfeld-nested-tires}); + \item \emph{tire graph} $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ + with outer/inner boundaries and annular edges + (\cite[Definition~1.5]{bauerfeld-nested-tires}); + \item \emph{partial tire dual} $D(T)$ + (\cite[Definition~1.7]{bauerfeld-nested-tires}); + \item face/edge counts + (\cite[Remark~1.6]{bauerfeld-nested-tires}). +\end{itemize} Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ and $G$ has $2n - 4$ triangular faces. -\begin{definition}[Level source] -A \emph{level source} of $G$ is any vertex $v \in V$; we write -$S = \{v\}$ for the level-0 source. -\end{definition} - -\begin{definition}[Levels] -Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is -$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest -source vertex. -\end{definition} - -\begin{definition}[Dual] -The \emph{dual} of $G$, written $G'$, is the inner (weak) planar dual of $G$ with -respect to the embedding $\Pi_G$: it has one vertex $d_f$ for each bounded face -$f$ of $G$, and an edge joining $d_f$ and $d_{f'}$ for each edge of $G$ shared by -two bounded faces $f$ and $f'$. The unbounded outer face contributes no vertex, -and edges of $G$ on the outer boundary contribute no dual edge. Since $G$ is a -triangulation, each vertex $d_f \in V(G')$ corresponds to a triangular face $f$ -of $G$, and we write $V(f) \subseteq V$ for its three incident vertices. -\end{definition} - -\begin{definition}[Dual depth] -\label{def:dual-depth} -Given a level source $S \subseteq V$, the \emph{dual depth} of a dual vertex -$d_f \in V(G')$ is -\[ - \delta_G(d_f) = \min_{v \in V(f)} \ell_G(v) - = \min_{v \in V(f)} \mathrm{dist}_G(v, S), -\] -the smallest level among the three vertices of $G$ bounding the face $f$. -\end{definition} - -\begin{figure}[h] -\centering -\includegraphics[width=0.7\textwidth]{fig_dual_depth.png} -\caption{Dual depth in a stacked-ring triangulation $G$ with level source -$S = \{0\}$. Each $G$ vertex is labelled by its level $\ell$. Each bounded face -carries a dual vertex (square, joined by dashed dual edges) coloured by its dual -depth $\delta(d_f) = \min_{v \in V(f)} \ell(v)$: the central fan has depth $0$, -the inner annulus depth $1$, and the outer annulus depth $2$. The outer face -(the level-$3$ triangle) is excluded from the inner dual and carries no dual -vertex.} -\label{fig:dual-depth} -\end{figure} - -\begin{definition}[Tire graph] -\label{def:tire-graph} -A \emph{tire graph} consists of a plane graph $T$ together with an -\emph{outer boundary} $B_{\mathrm{out}} \subseteq T$ and an \emph{inner -outerplanar graph} $O \subseteq T$ with $V(B_{\mathrm{out}}) \cap V(O) -= \emptyset$, where -\begin{itemize} - \item $B_{\mathrm{out}}$ is either a simple cycle of length $\geq 3$ - or a single vertex (a \emph{degenerate outer boundary}); - \item $O$ is an outerplanar graph; its \emph{inner boundary} - $B_{\mathrm{in}}$ is the closed walk in $O$ that traces the - boundary of $O$'s outer face in the inherited embedding, - which is a simple cycle when $O$ is $2$-connected and a - non-simple closed walk in general (visiting bridges twice and - cut-vertices multiple times); if $|V(O)| = 1$, we say $T$ has - a \emph{degenerate inner boundary}. -\end{itemize} -At most one of $B_{\mathrm{out}}, B_{\mathrm{in}}$ may be degenerate. -The vertex and edge sets of $T$ are -\[ - V(T) = V(B_{\mathrm{out}}) \cup V(O), - \qquad - E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}}, -\] -where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the -property that, in the plane embedding of $T$, the closed planar region -$R$ bounded externally by $B_{\mathrm{out}}$ and internally by -$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose -union is $R$. - -When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected, -$R$ is a closed annulus. More generally, $R$ is a closed planar -region that may fail to be a $2$-manifold at cut-vertices of $O$ (where -two ``lobes'' of the depth-$d$ region meet at a single vertex); the -inner boundary $B_{\mathrm{in}}$ is then a non-simple closed walk that -visits the cut-vertex multiple times. The relaxed definition -accommodates outerplanar inner graphs with bridges, cut-vertices, or -multiple connected components. When either boundary is degenerate, -$R$ is a closed disk with that vertex as apex. -\end{definition} - -\begin{figure}[h] -\centering -\includegraphics[width=0.78\textwidth]{fig_tire_example.png} -\caption{A tire graph with non-degenerate boundaries: outer boundary -$B_{\mathrm{out}}$ a $6$-cycle on vertices $0,\dots,5$ (blue), inner -boundary $B_{\mathrm{in}}$ a $4$-cycle on vertices $6,\dots,9$ (red), -inner outerplanar graph $O = B_{\mathrm{in}} \cup \{7\text{--}9\}$ -(with one chord, orange), and $E_{\mathrm{ann}}$ (grey) tiling the -annulus between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$ by ten -triangular faces.} -\label{fig:tire-example} -\end{figure} - -\begin{remark} -\label{rem:tire-counts} -Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By -Euler's formula on the annular (resp.\ disk) region $R$, the tire graph -has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$ -annular edges when neither boundary is degenerate; when exactly one -boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$ -triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$. -\end{remark} - -\begin{definition}[Partial tire dual] -\label{def:partial-tire-dual} -Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in -the sense of Definition~\ref{def:tire-graph}, and let $F_{\mathrm{ann}}$ -denote the set of triangular faces of $T$ in the closed annular region -between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$. The \emph{partial -tire dual} of $T$, written $D(T)$, is the graph defined as follows. - -\emph{Vertices.} -\begin{enumerate} - \item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an - \emph{interior vertex} $d_f$ of $D(T)$. - \item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a - \emph{leaf vertex} $\ell_e^{\mathrm{out}}$. - \item[(V3)] For each occurrence of an edge in the closed walk - $B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$), - a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$. (When $O$ is - $2$-connected each edge appears once; cut-vertices and - bridges of $O$ may cause an edge or vertex to appear more - than once.) -\end{enumerate} - -\emph{Edges.} -\begin{enumerate} - \item[(E1)] For each edge $e \in E(T)$ whose two incident faces both - lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}), - one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where - $f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces - incident to $e$. - \item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge - $\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where - $f \in F_{\mathrm{ann}}$ is the unique annular face incident - to $e$. The leaf $\ell_e^{\mathrm{out}}$ has degree $1$. - \item[(E3)] For each occurrence of $e$ on the boundary walk - $B_{\mathrm{in}}$, one edge - $\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where - $f \in F_{\mathrm{ann}}$ is the annular face incident to $e$ - on the side of that occurrence. The leaf - $\ell_e^{\mathrm{in}}$ has degree $1$. -\end{enumerate} -\end{definition} - -\begin{figure}[h] -\centering -\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual.png} -\caption{The partial tire dual $D(T)$ (purple squares + orange -diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ -and $k = 4$. The ten interior vertices $d_f$ at the centroids of the -annular triangles form a single $10$-cycle (solid purple); each -boundary edge of the annular region (either of $B_{\mathrm{out}}$ or -of $B_{\mathrm{in}}$) contributes a degree-$1$ leaf (orange diamond) -attached to the unique annular face incident to it (dashed orange), -giving the structure $C_{10} \circ K_1$ of -Proposition~\ref{prop:partial-tire-dual-structure}.} -\label{fig:partial-tire-dual-example} -\end{figure} - -\begin{figure}[h] -\centering -\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png} -\caption{Partial tire dual $D(T)$ when the inner outerplanar graph -$O$ has a bridge --- here a non-trivial edge cut connecting two -disjoint triangles. $B_{\mathrm{out}}$ is a $4$-cycle on -$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together -with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing -the bridge disconnects $O$). Because both faces incident to the -bridge are annular triangles, the bridge contributes an -\emph{interior dual edge} (highlighted in red) rather than two -leaves; consequently the interior dual subgraph is no longer the -single $(n+m)$-cycle of -Proposition~\ref{prop:partial-tire-dual-structure}, but a theta -graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident -annular faces) are joined by three internally vertex-disjoint paths -in $D(T)$. Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves) -and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves, -three for each triangle).} -\label{fig:partial-tire-dual-bridge} -\end{figure} \begin{proposition}[Structure of $D(T)$ when the annular triangulation is spoke-only] @@ -288,7 +127,7 @@ So each $d_f$ has degree $3$ in $D(T)$: two from interior edges (= spokes shared with adjacent annular faces) and one leaf. The induced subgraph on $\{d_f : f \in F_{\mathrm{ann}}\}$ is $2$-regular; together with the connectedness of the annular region this forces it to be a single -cycle. By Remark~\ref{rem:tire-counts}, the cycle has length $n + m$, +cycle. By \cite[Remark~1.6]{bauerfeld-nested-tires}, the cycle has length $n + m$, and there are also $n + m$ leaves attached one-per-cycle-vertex. \end{proof} @@ -371,7 +210,7 @@ its embedding from $\Pi_G$. Set $R_{C'} := \bigcup_{f \in F_{C'}} f \subseteq |\Pi_G|$. Then $C$, with the inherited embedding, is a tire graph in the sense of -Definition~\ref{def:tire-graph}. Its outer boundary +\cite[Definition~1.5]{bauerfeld-nested-tires}. Its outer boundary $B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$, namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$ (a simple cycle or single vertex); its inner outerplanar graph is $O = G[V_{C'} \cap @@ -434,7 +273,7 @@ cycle. At vertices $v \in L_{d+1} \cap V_{C'}$ the depth-$d$ faces may split into multiple arcs of $v$'s rotation; this corresponds exactly to $v$ being a cut-vertex of $O$, and the inner-side boundary walk visits $v$ correspondingly many times --- which is -already accommodated by Definition~\ref{def:tire-graph} (where +already accommodated by \cite[Definition~1.5]{bauerfeld-nested-tires} (where $B_{\mathrm{in}}$ is the outer-face boundary closed walk of $O$, not necessarily a simple cycle). @@ -487,7 +326,7 @@ the level-$D_{\max}$ cycle as the outer boundary. \label{rem:tire-no-extra-hypotheses} Two structural features of $R_{C'}$ that might at first appear to obstruct the tire-graph conclusion are both already accommodated by -Definition~\ref{def:tire-graph}: +\cite[Definition~1.5]{bauerfeld-nested-tires}: \emph{Cut-vertices of $O$.} A vertex $v \in V_{C'} \cap L_{d+1}$ may have the faces of $F_{C'}$ incident to it split into two or more @@ -574,9 +413,9 @@ its attached interior vertex. \begin{definition}[Tire annular subgraph] \label{def:tire-annular-subgraph} Let $G$ be a maximal planar graph with embedding $\Pi_G$ and inner -planar dual $G'$ (as in Definition~\ref{def:dual} above). Let +planar dual $G'$ (as in \cite[Definition~1.3]{bauerfeld-nested-tires} above). Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}}) \subseteq G$ be a tire -graph (Definition~\ref{def:tire-graph}), and let +graph (\cite[Definition~1.5]{bauerfeld-nested-tires}), and let $F_{\mathrm{ann}} \subseteq F(G)$ denote its set of annular faces. The \emph{tire annular subgraph} of $T$ in $G'$ is \[ @@ -760,6 +599,11 @@ E.~Bauerfeld, \emph{Plane Depth}, manuscript (math-research repository), 2026. +\bibitem{bauerfeld-nested-tires} +E.~Bauerfeld, +\emph{Coloring Nested Tire Graphs}, +manuscript (math-research repository), 2026. + \end{thebibliography} \end{document} diff --git a/papers/coloring_nested_tire_graphs/fig_dual_depth.png b/papers/coloring_nested_tire_graphs/fig_dual_depth.png new file mode 100644 index 0000000..fc1144b Binary files /dev/null and b/papers/coloring_nested_tire_graphs/fig_dual_depth.png differ diff --git a/papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png b/papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png new file mode 100644 index 0000000..3130934 Binary files /dev/null and b/papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png differ diff --git a/papers/coloring_nested_tire_graphs/fig_partial_tire_dual_bridge.png b/papers/coloring_nested_tire_graphs/fig_partial_tire_dual_bridge.png new file mode 100644 index 0000000..fa86ac1 Binary files /dev/null and b/papers/coloring_nested_tire_graphs/fig_partial_tire_dual_bridge.png differ diff --git a/papers/coloring_nested_tire_graphs/fig_tire_example.png b/papers/coloring_nested_tire_graphs/fig_tire_example.png new file mode 100644 index 0000000..c4f4316 Binary files /dev/null and b/papers/coloring_nested_tire_graphs/fig_tire_example.png differ diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux new file mode 100644 index 0000000..3d6054f --- /dev/null +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -0,0 +1,26 @@ +\relax +\citation{bauerfeld-nested-tire-duals} +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } +\newlabel{def:dual}{{1.3}{1}} +\newlabel{def:dual-depth}{{1.4}{2}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent } +\newlabel{fig:dual-depth}{{1}{2}} +\newlabel{def:tire-graph}{{1.5}{2}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent } +\newlabel{fig:tire-example}{{2}{3}} +\newlabel{rem:tire-counts}{{1.6}{3}} +\newlabel{def:partial-tire-dual}{{1.7}{3}} +\citation{bauerfeld-nested-tire-duals} +\bibcite{bauerfeld-depth}{1} +\bibcite{bauerfeld-nested-tire-duals}{2} +\newlabel{tocindent-1}{0pt} +\newlabel{tocindent0}{12.7778pt} +\newlabel{tocindent1}{17.77782pt} +\newlabel{tocindent2}{0pt} +\newlabel{tocindent3}{0pt} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ analysed in the companion paper\nonbreakingspace \cite {bauerfeld-nested-tire-duals}.}}{4}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-example}{{3}{4}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of the spoke-only case, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{5}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-bridge}{{4}{5}} +\gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log new file mode 100644 index 0000000..d625a80 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -0,0 +1,255 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:54 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**paper.tex +(./paper.tex +LaTeX2e <2021-11-15> patch level 1 +L3 programming layer <2022-02-24> +(/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls +Document Class: amsart 2020/05/29 v2.20.6 +\linespacing=\dimen138 +\normalparindent=\dimen139 +\normaltopskip=\skip47 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+\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{conjecture}[theorem]{Conjecture} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\newtheorem{xca}[theorem]{Exercise} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\numberwithin{equation}{section} + +\begin{document} + +\title{Coloring Nested Tire Graphs} + +% author one information +\author{Eric Bauerfeld} +\address{} +\curraddr{} +\email{} +\thanks{} + +\subjclass[2010]{Primary } + +\keywords{plane graph, triangulation, plane depth, level edge, dual graph, tire graph} + +\date{} + +\dedicatory{} + +\begin{abstract} +We establish the foundational definitions for studying the +Four Colour Theorem through nested level-structures on plane +triangulations. A \emph{level source} of a triangulation $G$ +induces a BFS layering of $G$, which in turn endows the inner +planar dual $G'$ with a \emph{dual depth} grading. We isolate the +basic object of study --- the \emph{tire graph} $T$, a plane graph +whose outer and inner boundaries bound an annular region +triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$ --- and +define its \emph{partial tire dual} $D(T)$, the dual restricted to +$T$'s annular faces together with leaves recording the boundary +edges. +\end{abstract} + +\maketitle + +\section{Introduction} + +A classical theorem of Tait recasts the Four Colour Theorem in dual, +edge-colouring terms: a plane triangulation $G$ is properly $4$-vertex-colourable +if and only if its dual cubic graph $G'$ is properly $3$-edge-colourable. Thus a +minimal counterexample to the Four Colour Theorem -- a smallest triangulation +admitting no proper $4$-colouring -- corresponds to a smallest cubic plane graph +admitting no proper $3$-edge-colouring. + +The structural study of such a minimal counterexample is the +overarching motivation for the present line of work. This first +paper establishes the foundational vocabulary --- level sources, +dual depth, tire graphs, and partial tire duals --- on which +subsequent papers in the series build. In particular, the +companion paper \cite{bauerfeld-nested-tire-duals} uses these +definitions to develop nested-cycle structure theorems and +chain-pigeonhole conjectures for tire annular subgraphs of $G'$. + +Throughout, $G = (V, E)$ is a plane maximal planar graph (a triangulation) +with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ +and $G$ has $2n - 4$ triangular faces. + +\begin{definition}[Level source] +A \emph{level source} of $G$ is any vertex $v \in V$; we write +$S = \{v\}$ for the level-0 source. +\end{definition} + +\begin{definition}[Levels] +Given a level source $S \subseteq V$, the \emph{level} of $v \in V$ is +$\ell_G(v) = \mathrm{dist}_G(v, S)$, the graph distance from $v$ to the nearest +source vertex. +\end{definition} + +\begin{definition}[Dual] +\label{def:dual} +The \emph{dual} of $G$, written $G'$, is the inner (weak) planar dual of $G$ with +respect to the embedding $\Pi_G$: it has one vertex $d_f$ for each bounded face +$f$ of $G$, and an edge joining $d_f$ and $d_{f'}$ for each edge of $G$ shared by +two bounded faces $f$ and $f'$. The unbounded outer face contributes no vertex, +and edges of $G$ on the outer boundary contribute no dual edge. Since $G$ is a +triangulation, each vertex $d_f \in V(G')$ corresponds to a triangular face $f$ +of $G$, and we write $V(f) \subseteq V$ for its three incident vertices. +\end{definition} + +\begin{definition}[Dual depth] +\label{def:dual-depth} +Given a level source $S \subseteq V$, the \emph{dual depth} of a dual vertex +$d_f \in V(G')$ is +\[ + \delta_G(d_f) = \min_{v \in V(f)} \ell_G(v) + = \min_{v \in V(f)} \mathrm{dist}_G(v, S), +\] +the smallest level among the three vertices of $G$ bounding the face $f$. +\end{definition} + +\begin{figure}[h] +\centering +\includegraphics[width=0.7\textwidth]{fig_dual_depth.png} +\caption{Dual depth in a stacked-ring triangulation $G$ with level source +$S = \{0\}$. Each $G$ vertex is labelled by its level $\ell$. Each bounded face +carries a dual vertex (square, joined by dashed dual edges) coloured by its dual +depth $\delta(d_f) = \min_{v \in V(f)} \ell(v)$: the central fan has depth $0$, +the inner annulus depth $1$, and the outer annulus depth $2$. The outer face +(the level-$3$ triangle) is excluded from the inner dual and carries no dual +vertex.} +\label{fig:dual-depth} +\end{figure} + +\begin{definition}[Tire graph] +\label{def:tire-graph} +A \emph{tire graph} consists of a plane graph $T$ together with an +\emph{outer boundary} $B_{\mathrm{out}} \subseteq T$ and an \emph{inner +outerplanar graph} $O \subseteq T$ with $V(B_{\mathrm{out}}) \cap V(O) += \emptyset$, where +\begin{itemize} + \item $B_{\mathrm{out}}$ is either a simple cycle of length $\geq 3$ + or a single vertex (a \emph{degenerate outer boundary}); + \item $O$ is an outerplanar graph; its \emph{inner boundary} + $B_{\mathrm{in}}$ is the closed walk in $O$ that traces the + boundary of $O$'s outer face in the inherited embedding, + which is a simple cycle when $O$ is $2$-connected and a + non-simple closed walk in general (visiting bridges twice and + cut-vertices multiple times); if $|V(O)| = 1$, we say $T$ has + a \emph{degenerate inner boundary}. +\end{itemize} +At most one of $B_{\mathrm{out}}, B_{\mathrm{in}}$ may be degenerate. +The vertex and edge sets of $T$ are +\[ + V(T) = V(B_{\mathrm{out}}) \cup V(O), + \qquad + E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}}, +\] +where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the +property that, in the plane embedding of $T$, the closed planar region +$R$ bounded externally by $B_{\mathrm{out}}$ and internally by +$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose +union is $R$. + +When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected, +$R$ is a closed annulus. More generally, $R$ is a closed planar +region that may fail to be a $2$-manifold at cut-vertices of $O$ (where +two ``lobes'' of the depth-$d$ region meet at a single vertex); the +inner boundary $B_{\mathrm{in}}$ is then a non-simple closed walk that +visits the cut-vertex multiple times. The relaxed definition +accommodates outerplanar inner graphs with bridges, cut-vertices, or +multiple connected components. When either boundary is degenerate, +$R$ is a closed disk with that vertex as apex. +\end{definition} + +\begin{figure}[h] +\centering +\includegraphics[width=0.78\textwidth]{fig_tire_example.png} +\caption{A tire graph with non-degenerate boundaries: outer boundary +$B_{\mathrm{out}}$ a $6$-cycle on vertices $0,\dots,5$ (blue), inner +boundary $B_{\mathrm{in}}$ a $4$-cycle on vertices $6,\dots,9$ (red), +inner outerplanar graph $O = B_{\mathrm{in}} \cup \{7\text{--}9\}$ +(with one chord, orange), and $E_{\mathrm{ann}}$ (grey) tiling the +annulus between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$ by ten +triangular faces.} +\label{fig:tire-example} +\end{figure} + +\begin{remark} +\label{rem:tire-counts} +Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By +Euler's formula on the annular (resp.\ disk) region $R$, the tire graph +has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$ +annular edges when neither boundary is degenerate; when exactly one +boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$ +triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$. +\end{remark} + +\begin{definition}[Partial tire dual] +\label{def:partial-tire-dual} +Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in +the sense of Definition~\ref{def:tire-graph}, and let $F_{\mathrm{ann}}$ +denote the set of triangular faces of $T$ in the closed annular region +between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$. The \emph{partial +tire dual} of $T$, written $D(T)$, is the graph defined as follows. + +\emph{Vertices.} +\begin{enumerate} + \item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an + \emph{interior vertex} $d_f$ of $D(T)$. + \item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a + \emph{leaf vertex} $\ell_e^{\mathrm{out}}$. + \item[(V3)] For each occurrence of an edge in the closed walk + $B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$), + a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$. (When $O$ is + $2$-connected each edge appears once; cut-vertices and + bridges of $O$ may cause an edge or vertex to appear more + than once.) +\end{enumerate} + +\emph{Edges.} +\begin{enumerate} + \item[(E1)] For each edge $e \in E(T)$ whose two incident faces both + lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}), + one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where + $f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces + incident to $e$. + \item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge + $\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where + $f \in F_{\mathrm{ann}}$ is the unique annular face incident + to $e$. The leaf $\ell_e^{\mathrm{out}}$ has degree $1$. + \item[(E3)] For each occurrence of $e$ on the boundary walk + $B_{\mathrm{in}}$, one edge + $\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where + $f \in F_{\mathrm{ann}}$ is the annular face incident to $e$ + on the side of that occurrence. The leaf + $\ell_e^{\mathrm{in}}$ has degree $1$. +\end{enumerate} +\end{definition} + +\begin{figure}[h] +\centering +\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual.png} +\caption{The partial tire dual $D(T)$ (purple squares + orange +diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ +and $k = 4$. The ten interior vertices $d_f$ at the centroids of the +annular triangles form a single $10$-cycle (solid purple); each +boundary edge of the annular region (either of $B_{\mathrm{out}}$ or +of $B_{\mathrm{in}}$) contributes a degree-$1$ leaf (orange diamond) +attached to the unique annular face incident to it (dashed orange), +giving the structure $C_{10} \circ K_1$ analysed in the companion +paper~\cite{bauerfeld-nested-tire-duals}.} +\label{fig:partial-tire-dual-example} +\end{figure} + +\begin{figure}[h] +\centering +\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png} +\caption{Partial tire dual $D(T)$ when the inner outerplanar graph +$O$ has a bridge --- here a non-trivial edge cut connecting two +disjoint triangles. $B_{\mathrm{out}}$ is a $4$-cycle on +$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together +with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing +the bridge disconnects $O$). Because both faces incident to the +bridge are annular triangles, the bridge contributes an +\emph{interior dual edge} (highlighted in red) rather than two +leaves; consequently the interior dual subgraph is no longer the +single $(n+m)$-cycle of the spoke-only case, but a theta +graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident +annular faces) are joined by three internally vertex-disjoint paths +in $D(T)$. Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves) +and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves, +three for each triangle).} +\label{fig:partial-tire-dual-bridge} +\end{figure} + +\begin{thebibliography}{9} + +\bibitem{bauerfeld-depth} +E.~Bauerfeld, +\emph{Plane Depth}, +manuscript (math-research repository), 2026. + +\bibitem{bauerfeld-nested-tire-duals} +E.~Bauerfeld, +\emph{Coloring Nested Tire Dual Graphs}, +manuscript (math-research repository), 2026. + +\end{thebibliography} + +\end{document}