coloring_nested_tire_graphs: add Definition 1.7 (Partial tire dual) + structure proposition

Adds Definition 1.7 (Partial tire dual) formalising the user's
construction: for a tire graph T with annular face set F_{ann}, the
partial tire dual D(T) has

  - Interior vertices d_f for each annular face f,
  - Leaf vertices for each edge of B_out and each occurrence of an
    edge on the boundary walk B_in (so cut-vertices/bridges of O
    contribute multiple leaves),
  - Interior dual edges for each annular edge incident to two annular
    faces,
  - Leaf edges from d_f to the corresponding leaf for each boundary
    edge of the annular region.

Adds Proposition 1.8 showing that when the annular triangulation is
spoke-only (i.e. every annular edge has one endpoint on B_out and one
on B_in) and O is 2-connected, each annular face has exactly 1
boundary edge + 2 interior annular edges.  Consequently each interior
vertex d_f has degree 3 = 2 (cycle) + 1 (leaf), and the induced
subgraph on {d_f} is a single cycle of length n + m.  D(T) is then
isomorphic to the corona C_{n+m} ∘ K_1 -- a cycle of length n+m with
one leaf attached to each cycle vertex; |V(D(T))| = |E(D(T))| = 2(n+m).

Subsequent numbering shifted: Proposition (Source-side simple-cycle
property) is now 1.9; Lemma (Tire-component) is now 1.10; Remarks
shift to 1.11 and 1.12.  All cross-references are by label, so they
update automatically.

Paper grows from 6 to 7 pages.

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

papers/coloring_nested_tire_graphs/paper.aux

@@ -7,17 +7,19 @@
 \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm  {out}}$ a $6$-cycle on vertices $0,\dots  ,5$ (blue), inner boundary $B_{\mathrm  {in}}$ a $4$-cycle on vertices $6,\dots  ,9$ (red), inner outerplanar graph $O = B_{\mathrm  {in}} \cup \{7\text  {--}9\}$ (with one chord, orange), and $E_{\mathrm  {ann}}$ (grey) tiling the annulus between $B_{\mathrm  {out}}$ and $B_{\mathrm  {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent }
 \newlabel{fig:tire-example}{{2}{3}}
 \newlabel{rem:tire-counts}{{1.6}{3}}
-\newlabel{prop:no-level-d-pinch}{{1.7}{3}}
+\newlabel{def:partial-tire-dual}{{1.7}{3}}
+\newlabel{prop:partial-tire-dual-structure}{{1.8}{4}}
+\newlabel{prop:no-level-d-pinch}{{1.9}{4}}
 \citation{bauerfeld-pds}
-\newlabel{lem:tire-component}{{1.8}{4}}
+\newlabel{lem:tire-component}{{1.10}{5}}
 \citation{bauerfeld-pds}
-\newlabel{rem:tire-component-degenerate}{{1.9}{5}}
+\newlabel{rem:tire-component-degenerate}{{1.11}{6}}
+\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{6}}
 \bibcite{bauerfeld-pds}{1}
 \newlabel{tocindent-1}{0pt}
 \newlabel{tocindent0}{12.7778pt}
 \newlabel{tocindent1}{17.77782pt}
 \newlabel{tocindent2}{0pt}
 \newlabel{tocindent3}{0pt}
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+\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{}\protected@file@percent }
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-\gdef \@abspage@last{6}

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papers/coloring_nested_tire_graphs/paper.tex

@@ -178,6 +178,82 @@ boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
 triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
 \end{remark}
 
+\begin{definition}[Partial tire dual]
+\label{def:partial-tire-dual}
+Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in
+the sense of Definition~\ref{def:tire-graph}, and let $F_{\mathrm{ann}}$
+denote the set of triangular faces of $T$ in the closed annular region
+between $B_{\mathrm{out}}$ and $B_{\mathrm{in}}$.  The \emph{partial
+tire dual} of $T$, written $D(T)$, is the graph defined as follows.
+
+\emph{Vertices.}
+\begin{enumerate}
+  \item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an
+        \emph{interior vertex} $d_f$ of $D(T)$.
+  \item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a
+        \emph{leaf vertex} $\ell_e^{\mathrm{out}}$.
+  \item[(V3)] For each occurrence of an edge in the closed walk
+        $B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$),
+        a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$.  (When $O$ is
+        $2$-connected each edge appears once; cut-vertices and
+        bridges of $O$ may cause an edge or vertex to appear more
+        than once.)
+\end{enumerate}
+
+\emph{Edges.}
+\begin{enumerate}
+  \item[(E1)] For each edge $e \in E(T)$ whose two incident faces both
+        lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}),
+        one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where
+        $f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces
+        incident to $e$.
+  \item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge
+        $\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where
+        $f \in F_{\mathrm{ann}}$ is the unique annular face incident
+        to $e$.  The leaf $\ell_e^{\mathrm{out}}$ has degree $1$.
+  \item[(E3)] For each occurrence of $e$ on the boundary walk
+        $B_{\mathrm{in}}$, one edge
+        $\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where
+        $f \in F_{\mathrm{ann}}$ is the annular face incident to $e$
+        on the side of that occurrence.  The leaf
+        $\ell_e^{\mathrm{in}}$ has degree $1$.
+\end{enumerate}
+\end{definition}
+
+\begin{proposition}[Structure of $D(T)$ when the annular triangulation
+is spoke-only]
+\label{prop:partial-tire-dual-structure}
+Suppose $B_{\mathrm{out}}$ is a simple cycle of length $n$, $O$ is a
+$2$-connected outerplanar graph whose outer-face cycle
+$B_{\mathrm{in}}$ has length $m$, and $E_{\mathrm{ann}}$ consists only
+of \emph{spokes} (edges with one endpoint in $V(B_{\mathrm{out}})$ and
+one in $V(B_{\mathrm{in}})$).  Then each face $f \in F_{\mathrm{ann}}$
+has exactly one boundary edge (on $B_{\mathrm{out}}$ or
+$B_{\mathrm{in}}$) and two interior annular edges, and consequently
+$D(T)$ is isomorphic to the corona graph $C_{n+m} \circ K_1$: a cycle
+of length $n + m$ on the interior vertices $\{d_f\}$, with one leaf
+attached to each cycle vertex.
+
+In particular, $|V(D(T))| = 2(n+m)$ and $|E(D(T))| = 2(n+m)$.
+\end{proposition}
+
+\begin{proof}
+Each annular triangle $f$ in a spoke-only triangulation has the form
+$\{x, y, z\}$ with $x \in V(B_{\mathrm{out}})$, $y \in V(B_{\mathrm{in}})$,
+and $z$ also in $V(B_{\mathrm{out}}) \cup V(B_{\mathrm{in}})$.  Of its
+three edges, the one between the two same-side vertices
+($x$-$z$ if both on $B_{\mathrm{out}}$, or $y$-$z$ if both on
+$B_{\mathrm{in}}$) is a boundary edge of the annular region; the
+other two edges are spokes.
+
+So each $d_f$ has degree $3$ in $D(T)$: two from interior edges (= spokes
+shared with adjacent annular faces) and one leaf.  The induced subgraph
+on $\{d_f : f \in F_{\mathrm{ann}}\}$ is $2$-regular; together with the
+connectedness of the annular region this forces it to be a single
+cycle.  By Remark~\ref{rem:tire-counts}, the cycle has length $n + m$,
+and there are also $n + m$ leaves attached one-per-cycle-vertex.
+\end{proof}
+
 \begin{proposition}[Source-side simple-cycle property]
 \label{prop:no-level-d-pinch}
 Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and