diff --git a/papers/coloring_nested_tire_dual_graphs/paper.aux b/papers/coloring_nested_tire_dual_graphs/paper.aux index 4696eed..77575ce 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.aux +++ b/papers/coloring_nested_tire_dual_graphs/paper.aux @@ -7,31 +7,34 @@ \citation{bauerfeld-nested-tires} \citation{bauerfeld-nested-tires} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } -\newlabel{prop:partial-tire-dual-structure}{{1.1}{1}} +\newlabel{def:partial-tire-dual}{{1.1}{1}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\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 1.2\hbox {}.}}{2}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-example}{{1}{2}} +\newlabel{prop:partial-tire-dual-structure}{{1.2}{2}} \citation{bauerfeld-nested-tires} -\newlabel{prop:no-level-d-pinch}{{1.2}{2}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\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 1.2\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).}}{3}{}\protected@file@percent } +\newlabel{fig:partial-tire-dual-bridge}{{2}{3}} +\newlabel{prop:no-level-d-pinch}{{1.3}{3}} +\citation{bauerfeld-nested-tires} +\newlabel{lem:tire-component}{{1.4}{4}} +\citation{bauerfeld-depth} \citation{bauerfeld-nested-tires} \citation{bauerfeld-depth} \citation{bauerfeld-nested-tires} -\newlabel{lem:tire-component}{{1.3}{3}} -\citation{bauerfeld-depth} -\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}} +\newlabel{rem:tire-component-degenerate}{{1.5}{6}} +\newlabel{rem:tire-no-extra-hypotheses}{{1.6}{6}} +\newlabel{prop:edge-vertex-bijection}{{1.7}{6}} \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}} +\newlabel{rem:edge-vertex-corollary}{{1.8}{7}} +\newlabel{def:tire-annular-subgraph}{{1.9}{7}} +\newlabel{def:tire-annular-face-connector}{{1.10}{7}} +\newlabel{def:spokes}{{1.11}{7}} +\newlabel{rem:facial-dual-spoke-only}{{1.12}{8}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\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 $u_4, u_5$).}}{8}{}\protected@file@percent } +\newlabel{fig:facial-dual-choices}{{3}{8}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{A conjectural Latin-style substructure}}{8}{}\protected@file@percent } +\newlabel{sec:latin-conjecture}{{2}{8}} \bibcite{bauerfeld-depth}{1} \bibcite{bauerfeld-nested-tires}{2} \newlabel{tocindent-1}{0pt} @@ -39,5 +42,7 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent } -\gdef \@abspage@last{8} +\newlabel{conj:latin}{{2.1}{9}} +\newlabel{conj:chain-latin}{{2.2}{9}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent } +\gdef \@abspage@last{9} diff --git a/papers/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index b9e1bdf..bd2c1c9 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 00:54 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:57 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,19 +192,31 @@ 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] [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 -495. -(pdftex.def) Requested size: 360.0pt x 143.50418pt. + +File: fig_partial_tire_dual.png Graphic file (type png) + +Package pdftex.def Info: fig_partial_tire_dual.png used on input line 143. +(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 1 +58. +(pdftex.def) Requested size: 306.0022pt x 204.59406pt. 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PDF statistics: - 144 PDF objects out of 1000 (max. 8388607) - 87 compressed objects within 1 object stream + 151 PDF objects out of 1000 (max. 8388607) + 89 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) - 6 words of extra memory for PDF output out of 10000 (max. 10000000) + 16 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 4c5e064..607c03a 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 2b4ac16..9f8ce8f 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.tex +++ b/papers/coloring_nested_tire_dual_graphs/paper.tex @@ -86,8 +86,6 @@ use, without restating, the notions of: \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} @@ -97,6 +95,86 @@ with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$ and $G$ has $2n - 4$ triangular faces. +\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 \cite[Definition~1.5]{bauerfeld-nested-tires}, 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] \label{prop:partial-tire-dual-structure} diff --git a/papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png b/papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png deleted file mode 100644 index 3130934..0000000 Binary files a/papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png and /dev/null 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 deleted file mode 100644 index fa86ac1..0000000 Binary files a/papers/coloring_nested_tire_graphs/fig_partial_tire_dual_bridge.png and /dev/null differ diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 3d6054f..ac01d65 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -6,11 +6,6 @@ \@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} @@ -18,9 +13,8 @@ \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} +\@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}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent } +\gdef \@abspage@last{3} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index d625a80..476f6a6 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 00:54 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:57 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -195,61 +195,44 @@ e File: fig_dual_depth.png Graphic file (type png) -Package pdftex.def Info: fig_dual_depth.png used on input line 119. +Package pdftex.def Info: fig_dual_depth.png used on input line 117. 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PDF statistics: - 109 PDF objects out of 1000 (max. 8388607) - 61 compressed objects within 1 object stream + 94 PDF objects out of 1000 (max. 8388607) + 54 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) - 21 words of extra memory for PDF output out of 10000 (max. 10000000) + 11 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 952d72e..6035c95 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 770660f..b7aabd2 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -52,9 +52,7 @@ 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. +record its face/edge counts. \end{abstract} \maketitle @@ -191,83 +189,6 @@ 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}