coloring_nested_tire_graphs: rename "partial tire facial dual" → "tire annular face connector"

Renames Definition 1.16 from "Partial tire facial dual" to "Tire
annular face connector" to match the family of "tire annular ..."
terminology (cf. Definition 1.15 "Tire annular subgraph").

Symbol T'_{f'} unchanged.

Updates:
- Definition 1.16 title and body wording
- Label changed to def:tire-annular-face-connector
- Figure suptitle in fig_facial_dual_choices.png
- Module docstring of draw_facial_dual_choices.py

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

@@ -1,5 +1,5 @@
 """For the bridge case where T'_ann = theta(1, p, q) has three faces,
-illustrate how the partial tire facial dual T'_{f'} (Definition 1.15)
+illustrate how the tire annular face connector T'_{f'} (Definition 1.16)
 depends on the choice of face f'.
 
 We use theta(1, 3, 3) = C_6 + chord (v_0, v_3) as the smallest
@@ -155,7 +155,7 @@ def main():
                show_external_for=ext_candidates,
                face_shade=shadeC)
 
-    fig.suptitle(r"Partial tire facial dual $T'_{f'}$ for the bridge case " +
+    fig.suptitle(r"Tire annular face connector $T'_{f'}$ for the bridge case " +
                  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 $v \in V(f')$.",

papers/coloring_nested_tire_graphs/paper.aux

@@ -28,7 +28,7 @@
 \newlabel{tocindent3}{0pt}
 \newlabel{rem:edge-vertex-corollary}{{1.14}{9}}
 \newlabel{def:tire-annular-subgraph}{{1.15}{9}}
-\newlabel{def:partial-tire-facial-dual}{{1.16}{9}}
+\newlabel{def:tire-annular-face-connector}{{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 }

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 23:06
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
@@ -244,7 +244,7 @@ LaTeX Warning: There were undefined references.
  ) 
 Here is how much of TeX's memory you used:
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- 43054 string characters out of 5846347
+ 43057 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
@@ -266,7 +266,7 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive
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-Output written on paper.pdf (10 pages, 813353 bytes).
+Output written on paper.pdf (10 pages, 813184 bytes).
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  73 compressed objects within 1 object stream

papers/coloring_nested_tire_graphs/paper.tex

@@ -587,14 +587,14 @@ 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}
+\begin{definition}[Tire annular face connector]
+\label{def:tire-annular-face-connector}
 With $G, G', T$ as in
 Definition~\ref{def:tire-annular-subgraph}, let $f'$ be a face of the
 tire annular subgraph $T'_{\mathrm{ann}}$ in its inherited embedding,
 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
+vertices on the boundary walk of $f'$.  The \emph{tire annular face
+connector 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)