diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 14180d5..9f155fe 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -27,5 +27,7 @@ \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} \newlabel{rem:edge-vertex-corollary}{{1.14}{9}} +\newlabel{def:partial-tire-facial-dual}{{1.15}{9}} +\newlabel{rem:facial-dual-spoke-only}{{1.16}{9}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent } \gdef \@abspage@last{9} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 187d619..ba18c94 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) 25 MAY 2026 20:52 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 22:36 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -224,33 +224,40 @@ Package pdftex.def Info: fig_partial_tire_dual_bridge.png used on input line 2 LaTeX Warning: `h' float specifier changed to `ht'. 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PDF statistics: 123 PDF objects out of 1000 (max. 8388607) 71 compressed objects within 1 object stream diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 0781e84..d872b53 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 1136158..5d75edd 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -569,6 +569,59 @@ itself; its color is freely determined as the missing third color at its attached interior vertex. \end{remark} +\begin{definition}[Partial tire facial dual] +\label{def:partial-tire-facial-dual} +Let $G$ be a maximal planar graph with embedding $\Pi_G$ and inner +planar dual $G'$ (as in Definition~\ref{def:dual} above). Let +$T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}}) \subseteq G$ be a tire +graph (Definition~\ref{def:tire-graph}), and let +$F_{\mathrm{ann}} \subseteq F(G)$ denote its set of annular faces. + +\smallskip +\noindent\textbf{(i) Annular dual subgraph.} Define +\[ + T'_{\mathrm{ann}} \;:=\; G'\bigl[\,\{d_f : f \in F_{\mathrm{ann}}\}\,\bigr], +\] +the subgraph of $G'$ induced on the dual vertices corresponding to the +annular faces of $T$. Equip $T'_{\mathrm{ann}}$ with the planar +embedding inherited from $G'$ (which, by deletion of vertices outside +the annulus, remains a planar embedding of $T'_{\mathrm{ann}}$ in the +sense of $\Pi_G$). + +\smallskip +\noindent\textbf{(ii) Partial tire facial dual.} For each face $f'$ +of $T'_{\mathrm{ann}}$ in its inherited embedding, let +$V(f') \subseteq V(T'_{\mathrm{ann}})$ denote the set of vertices on +the boundary walk of $f'$. Define the \emph{partial tire facial +dual at $f'$} to be the subgraph +\[ + T'_{f'} \;:=\; \bigl(\,V(f') \cup N_{G'}(V(f'))\,,\; + \{\,e \in E(G') : e \text{ is incident to } V(f')\,\}\,\bigr) + \;\subseteq\; G', +\] +i.e.\ the subgraph of $G'$ on the closed $G'$-neighborhood of $V(f')$ +together with every $G'$-edge incident to $V(f')$. +\end{definition} + +\begin{remark} +\label{rem:facial-dual-spoke-only} +In the spoke-only setting of +Proposition~\ref{prop:partial-tire-dual-structure}, the annular +dual subgraph is $T'_{\mathrm{ann}} = \Gamma \cong C_{n+m}$ +(Proposition~\ref{prop:edge-vertex-bijection}). This cycle has exactly +two faces in its inherited embedding -- one on each side of the cycle +in $\Pi_G$ -- and both face boundaries traverse all $n+m$ vertices, so +$V(f') = V(\Gamma)$ for either choice of $f'$. Each interior dual +vertex $d_f$ has $G'$-degree $3$ (since $G$ is a triangulation), of +which two edges lie in $\Gamma$ (cycle edges) and one edge points to a +single non-annular face of $G$. Consequently $T'_{f'}$ has $n + m$ +interior vertices plus the non-annular face vertices to which they +connect, and is independent of the choice of $f'$. When $G$ consists +only of the tire $T$ together with one source-side face inside +$B_{\mathrm{out}}$ and one $O$-side face inside $B_{\mathrm{in}}$, +$T'_{f'}$ recovers the planar dual of $T$ itself. +\end{remark} + \begin{thebibliography}{9} \bibitem{bauerfeld-pds}