coloring_nested_tire_graphs: add Definition 1.15 (partial tire facial dual)

Adds a new definition formalizing the "partial tire facial dual" T'_{f'}:

  (i) Annular dual subgraph T'_ann := G'[{d_f : f ∈ F_ann}], with
      planar embedding inherited from G' (where G' is the inner
      planar dual of the maximal planar G).

  (ii) For each face f' of T'_ann in its inherited embedding,
       T'_{f'} := closed G'-neighborhood of V(f') together with
       every G'-edge incident to V(f').

Adds a remark noting that in the spoke-only case T'_ann = Γ ≅ C_{n+m}
has two faces (both with V(f') = all interior dual vertices), and
T'_{f'} recovers the planar dual of T when G is the tire plus one
source-side and one O-side face.

Paper stays at 9 pages.

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

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}

papers/coloring_nested_tire_graphs/paper.log

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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}