coloring_nested_tire_graphs: rename "annular dual subgraph" → "tire annular subgraph"; revert symbol to T'_ann

Revert the previous renaming of G'_ann; the symbol is back to
T'_ann.  The CONCEPT NAME is changed from "annular dual subgraph"
to "tire annular subgraph" to clarify it's the tire's annular
portion as seen in G'.

Updates:
- Definition 1.15 retitled "Tire annular subgraph"
- Label changed to def:tire-annular-subgraph
- Cross-references in Definition 1.16 and the spoke-only remark
- Figure suptitle reverted to T'_ann
- Regenerated 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"($G'_{\mathrm{ann}} = \theta(1,3,3)$, three faces $A,B,C$)" + "\n" +
+                 r"($T'_{\mathrm{ann}} = \theta(1,3,3)$, 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

@@ -27,10 +27,10 @@
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
 \newlabel{rem:edge-vertex-corollary}{{1.14}{9}}
-\newlabel{def:annular-dual-subgraph}{{1.15}{9}}
+\newlabel{def:tire-annular-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: $G'_{\mathrm  {ann}} = \theta (1, 3, 3)$ 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 }
+\@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}}
 \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 22:49
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+Output written on paper.pdf (10 pages, 813353 bytes).
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papers/coloring_nested_tire_graphs/paper.tex

@@ -569,30 +569,30 @@ itself; its color is freely determined as the missing third color at
 its attached interior vertex.
 \end{remark}
 
-\begin{definition}[Annular dual subgraph]
+\begin{definition}[Tire annular subgraph]
-\label{def:annular-dual-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
 $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}}) \subseteq G$ be a tire
 graph (Definition~\ref{def:tire-graph}), and let
 $F_{\mathrm{ann}} \subseteq F(G)$ denote its set of annular faces.
-The \emph{annular dual subgraph} of $T$ in $G'$ is
+The \emph{tire annular subgraph} of $T$ in $G'$ is
 \[
-  G'_{\mathrm{ann}} \;:=\; G'\bigl[\,\{d_f : f \in F_{\mathrm{ann}}\}\,\bigr],
+  T'_{\mathrm{ann}} \;:=\; G'\bigl[\,\{d_f : f \in F_{\mathrm{ann}}\}\,\bigr],
 \]
 the subgraph of $G'$ induced on the dual vertices corresponding to the
-annular faces of $T$.  We equip $G'_{\mathrm{ann}}$ with the planar
+annular faces of $T$.  We equip $T'_{\mathrm{ann}}$ with the planar
 embedding inherited from $G'$ (which, by deletion of vertices outside
-the annulus, remains a planar embedding of $G'_{\mathrm{ann}}$ in the
+the annulus, remains a planar embedding of $T'_{\mathrm{ann}}$ in the
 sense of $\Pi_G$).
 \end{definition}
 
 \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
+Definition~\ref{def:tire-annular-subgraph}, let $f'$ be a face of the
-annular dual subgraph $G'_{\mathrm{ann}}$ in its inherited embedding,
+tire annular subgraph $T'_{\mathrm{ann}}$ in its inherited embedding,
-and let $V(f') \subseteq V(G'_{\mathrm{ann}})$ denote the set of
+and let $V(f') \subseteq V(T'_{\mathrm{ann}})$ denote the set of
 vertices on the boundary walk of $f'$.  The \emph{partial tire facial
 dual at $f'$} is the subgraph
 \[
@@ -607,8 +607,8 @@ together with every $G'$-edge incident to $V(f')$.
 \begin{remark}
 \label{rem:facial-dual-spoke-only}
 In the spoke-only setting of
-Proposition~\ref{prop:partial-tire-dual-structure}, the annular
+Proposition~\ref{prop:partial-tire-dual-structure}, the tire annular
-dual subgraph is $G'_{\mathrm{ann}} = \Gamma \cong C_{n+m}$
+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
@@ -627,16 +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: $G'_{\mathrm{ann}} = \theta(1, 3, 3)$ has three faces
+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 $G'_{\mathrm{ann}}$,
+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 $G'_{\mathrm{ann}}$,
+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 $u_1, u_2$