diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py b/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py index 844e44c..17ce3ab 100644 --- a/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py +++ b/papers/coloring_nested_tire_graphs/experiments/draw_facial_dual_choices.py @@ -156,7 +156,7 @@ def main(): face_shade=shadeC) fig.suptitle(r"Partial tire facial dual $T'_{f'}$ for the bridge case " + - r"($T'_{\mathrm{ann}} = \theta(1,3,3)$, three faces $A,B,C$)" + "\n" + + r"($G'_{\mathrm{ann}} = \theta(1,3,3)$, three faces $A,B,C$)" + "\n" + r"Blue: edges of $T'_{f'}$. Dark circles: $V(f')$. " + r"Red squares: external $G'$-neighbors $u_v$ included via $v \in V(f')$.", fontsize=11, y=1.02) diff --git a/papers/coloring_nested_tire_graphs/notes/fig_facial_dual_choices.png b/papers/coloring_nested_tire_graphs/notes/fig_facial_dual_choices.png index 8ec54d9..d8e370f 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/fig_facial_dual_choices.png and b/papers/coloring_nested_tire_graphs/notes/fig_facial_dual_choices.png differ diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 1128855..a12e01e 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -27,9 +27,10 @@ \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}} +\newlabel{def:annular-dual-subgraph}{{1.15}{9}} +\newlabel{def:partial-tire-facial-dual}{{1.16}{9}} +\newlabel{rem:facial-dual-spoke-only}{{1.17}{9}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent } -\@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 } +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The bridge case: $G'_{\mathrm {ann}} = \theta (1, 3, 3)$ 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 $G'_{\mathrm {ann}}$, 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 $G'_{\mathrm {ann}}$, 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}} \gdef \@abspage@last{10} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 45a1d1e..bdea733 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 22:42 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 22:46 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -232,7 +232,7 @@ LaTeX Warning: Reference `def:dual' on page 9 undefined on input line 575. 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 -627. +628. (pdftex.def) Requested size: 360.0pt x 143.50418pt. LaTeX Warning: `h' float specifier changed to `ht'. @@ -243,10 +243,10 @@ LaTeX Warning: There were undefined references. ) Here is how much of TeX's memory you used: - 3040 strings out of 478268 - 43027 string characters out of 5846347 - 344270 words of memory out of 5000000 - 21083 multiletter control sequences out of 15000+600000 + 3041 strings out of 478268 + 43054 string characters out of 5846347 + 344281 words of memory out of 5000000 + 21084 multiletter control sequences out of 15000+600000 475666 words of font info for 53 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 69i,8n,76p,1079b,316s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -266,7 +266,7 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb> -Output written on paper.pdf (10 pages, 814715 bytes). +Output written on paper.pdf (10 pages, 813723 bytes). PDF statistics: 128 PDF objects out of 1000 (max. 8388607) 73 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 0b521c8..a3276c6 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 c407ce3..cb4ff55 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -569,31 +569,32 @@ 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} +\begin{definition}[Annular dual subgraph] +\label{def:annular-dual-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 $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}}) \subseteq G$ 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 +The \emph{annular dual subgraph} of $T$ in $G'$ is \[ - T'_{\mathrm{ann}} \;:=\; G'\bigl[\,\{d_f : f \in F_{\mathrm{ann}}\}\,\bigr], + G'_{\mathrm{ann}} \;:=\; G'\bigl[\,\{d_f : f \in F_{\mathrm{ann}}\}\,\bigr], \] the subgraph of $G'$ induced on the dual vertices corresponding to the -annular faces of $T$. Equip $T'_{\mathrm{ann}}$ with the planar +annular faces of $T$. We equip $G'_{\mathrm{ann}}$ 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 +the annulus, remains a planar embedding of $G'_{\mathrm{ann}}$ in the sense of $\Pi_G$). +\end{definition} -\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 +\begin{definition}[Partial tire facial dual] +\label{def:partial-tire-facial-dual} +With $G, G', T$ as in +Definition~\ref{def:annular-dual-subgraph}, let $f'$ be a face of the +annular dual subgraph $G'_{\mathrm{ann}}$ in its inherited embedding, +and let $V(f') \subseteq V(G'_{\mathrm{ann}})$ denote the set of +vertices on the boundary walk of $f'$. The \emph{partial tire facial +dual at $f'$} is the subgraph \[ T'_{f'} \;:=\; \bigl(\,V(f') \cup N_{G'}(V(f'))\,,\; \{\,e \in E(G') : e \text{ is incident to } V(f')\,\}\,\bigr) @@ -607,7 +608,7 @@ together with every $G'$-edge incident to $V(f')$. \label{rem:facial-dual-spoke-only} In the spoke-only setting of Proposition~\ref{prop:partial-tire-dual-structure}, the annular -dual subgraph is $T'_{\mathrm{ann}} = \Gamma \cong C_{n+m}$ +dual subgraph is $G'_{\mathrm{ann}} = \Gamma \cong C_{n+m}$ (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 @@ -626,16 +627,16 @@ $T'_{f'}$ recovers the planar dual of $T$ itself. \centering \includegraphics[width=\textwidth]{notes/fig_facial_dual_choices.png} \caption{\label{fig:facial-dual-choices} -The bridge case: $T'_{\mathrm{ann}} = \theta(1, 3, 3)$ has three faces +The bridge case: $G'_{\mathrm{ann}} = \theta(1, 3, 3)$ 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 $T'_{\mathrm{ann}}$, +the bridge edge) have all three $G'$-edges inside $G'_{\mathrm{ann}}$, 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 $T'_{\mathrm{ann}}$, +gray circles are $G'$-neighbors of $V(f')$ within $G'_{\mathrm{ann}}$, 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 $u_1, u_2$