coloring_nested_tire_graphs: prove Lemma 1.7 using Lemma 2.6 of plane_depth_sequencing

- Adds a proof sketch to Lemma 1.7 (tire-component lemma).  The
  outerplanarity step cites Lemma 2.6 of `bauerfeld-pds` (the
  Plane Depth Sequencing manuscript), which proves that for any
  level source the subgraph induced on a single depth level is
  outerplanar.  The proof notes that the cited result, originally
  stated for an outer-cycle source, extends verbatim to an arbitrary
  level source by treating S as the depth-0 set.
- The remainder of the proof: layer containment forces V_{C'} ⊆
  L_d ∪ L_{d+1}; a face-by-face boundary analysis shows ∂R_{C'} is
  monochromatic in level; connectivity of C' rules out higher genus;
  the resulting one or two closed boundary walks give the tire
  graph's two boundary parts (with the degenerate case at the BFS
  endpoints).
- Adds a thebibliography block with one entry for the cited paper.

The paper grows from 3 to 4 pages.

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

papers/coloring_nested_tire_graphs/paper.aux

@@ -5,13 +5,16 @@
 \newlabel{fig:dual-depth}{{1}{2}}
 \newlabel{def:tire-graph}{{1.5}{2}}
 \newlabel{rem:tire-counts}{{1.6}{2}}
-\newlabel{tocindent-1}{0pt}
+\citation{bauerfeld-pds}
-\newlabel{tocindent0}{0pt}
-\newlabel{tocindent1}{17.77782pt}
-\newlabel{tocindent2}{0pt}
-\newlabel{tocindent3}{0pt}
 \@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{lem:tire-component}{{1.7}{3}}
-\newlabel{rem:tire-component-degenerate}{{1.8}{3}}
+\bibcite{bauerfeld-pds}{1}
-\gdef \@abspage@last{3}
+\newlabel{tocindent-1}{0pt}
+\newlabel{tocindent0}{12.7778pt}
+\newlabel{tocindent1}{17.77782pt}
+\newlabel{tocindent2}{0pt}
+\newlabel{tocindent3}{0pt}
+\newlabel{rem:tire-component-degenerate}{{1.8}{4}}
+\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
+\gdef \@abspage@last{4}

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papers/coloring_nested_tire_graphs/paper.tex

@@ -194,6 +194,47 @@ $C$ inside its closed boundary region are exactly the faces of $G$ in
 $F_{C'}$.
 \end{lemma}
 
+\begin{proof}[Proof sketch]
+By Lemma~2.6 of \cite{bauerfeld-pds} (whose argument, given for an
+outer-cycle source, extends verbatim to an arbitrary level source $S$
+by treating $S$ as the depth-$0$ set), the subgraph $G[L_{d'}]$ is
+outerplanar for each $d' \geq 0$.  Since subgraphs of outerplanar
+graphs are outerplanar, both $G[V_{C'} \cap L_d]$ and
+$G[V_{C'} \cap L_{d+1}]$ are outerplanar.
+
+Layer containment (a consequence of the bounded-step property of BFS
+on a triangulation) gives $V_{C'} \subseteq L_d \cup L_{d+1}$, so $C$
+has vertex partition $V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap
+L_{d+1})$, and every face $f \in F_{C'}$ is a triangle with at least
+one vertex in $L_d$.
+
+It remains to identify the boundary of $R_{C'} := \bigcup_{f \in
+F_{C'}} f \subseteq |\Pi_G|$.  Each edge on the topological boundary
+$\partial R_{C'}$ separates a face $f \in F_{C'}$ (depth $d$) from a
+face $f' \notin F_{C'}$.  Such an $f'$ has dual depth in $\{d-1, d+1\}$:
+if $d$, then $f'$ shares an edge with a depth-$d$ face but lies in a
+distinct component of $G'_d$, which contradicts the connectivity of
+$C'$ together with the fact that adjacent depth-$d$ dual vertices belong
+to the same component.  A short case analysis on the level of the third
+vertex of $f'$ shows that boundary edges with $\delta(f') = d-1$ have
+both endpoints in $L_d$, while those with $\delta(f') = d+1$ have both
+endpoints in $L_{d+1}$.  Each connected component of $\partial R_{C'}$
+is therefore monochromatic in level.
+
+Since $C'$ is connected and $R_{C'}$ is a connected planar 2-complex
+of triangles glued along edges, $\partial R_{C'}$ consists of either
+one closed walk (when $R_{C'}$ is a topological disk) or two closed
+walks (when $R_{C'}$ is a topological annulus).  These walks are
+simple cycles in $G$ on the respective level sets (with one cycle
+possibly degenerating to a single vertex of $L_d$ or $L_{d+1}$ at the
+endpoints of the BFS, giving the degenerate-boundary case of
+Definition~\ref{def:tire-graph}).  Together with the outerplanarity of
+$G[V_{C'} \cap L_d]$ and $G[V_{C'} \cap L_{d+1}]$ established above,
+these cycles serve as the two boundary parts of $C$, in either order,
+and the depth-$d$ triangles in $F_{C'}$ tile the closed region between
+them.
+\end{proof}
+
 \begin{remark}
 \label{rem:tire-component-degenerate}
 Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be
@@ -206,4 +247,13 @@ the orientation of the inherited embedding (equivalently, on which side
 of $C$ contains the rest of $\Pi_G$).
 \end{remark}
 
+\begin{thebibliography}{9}
+
+\bibitem{bauerfeld-pds}
+E.~Bauerfeld,
+\emph{Plane Depth Sequencing},
+manuscript (math-research repository), 2026.
+
+\end{thebibliography}
+
 \end{document}