coloring_nested_tire_graphs: split annular dual subgraph into its own definition; rename T'_ann → G'_ann

Splits the old Definition 1.15 (which combined the annular dual
subgraph and the partial tire facial dual) into two separate
definitions:

  Definition 1.15 (Annular dual subgraph):
    G'_ann := G'[{d_f : f ∈ F_ann}], with planar embedding inherited
    from G'.  Renamed from T'_ann to G'_ann (since it's an induced
    subgraph of G', not of T).

  Definition 1.16 (Partial tire facial dual):
    T'_{f'} := closed G'-neighborhood of V(f') with every G'-edge
    incident to V(f'), for f' a face of G'_ann.

Updates all references in paper.tex and in the Figure 5 caption /
figure title; regenerates fig_facial_dual_choices.png.

Paper stays at 10 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

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)

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}

papers/coloring_nested_tire_graphs/paper.log

@@ -1,4 +1,4 @@
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+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
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@@ -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)
 <use notes/fig_facial_dual_choices.png>
 Package pdftex.def Info: notes/fig_facial_dual_choices.png  used on input line 
-627.
+628.
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 LaTeX Warning: `h' float specifier changed to `ht'.
@@ -243,10 +243,10 @@ LaTeX Warning: There were undefined references.
 
  ) 
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+Output written on paper.pdf (10 pages, 813723 bytes).
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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 $u_1, u_2$