coloring_nested_tire_graphs: rename induced tire graph C -> T_{C'}

In the tire-component lemma the induced subgraph that becomes the tire
graph was named C, clashing with C used everywhere else for cycles
(seam cycles C_T, cycle graphs C_n, the seam cycle C in Def 1.16).
Rename it to T_{C'} throughout the lemma statement, its proof, and the
degenerate-boundary remark, so C/C'/C_T are uniformly reserved for
cycles and components.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

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)  1 JUN 2026 00:17
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@@ -546,7 +546,7 @@ mr/m/it/10 has $\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/10 (\OML/cmm/m
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+Output written on paper.pdf (16 pages, 929386 bytes).
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papers/coloring_nested_tire_graphs/paper.tex

@@ -298,17 +298,17 @@ outer face (such an embedding exists for any planar graph and any
 single-vertex source).  For $d \geq 0$, let $C'$ be a connected
 component of the depth-$d$ dual subgraph $G'_d$, with faces $F_{C'}$,
 bounding vertices $V_{C'}$, and region $R_{C'}$ as in
-Definition~\ref{def:dual-component}; let $C := G[V_{C'}]$ inherit its
+Definition~\ref{def:dual-component}; let $T_{C'} := G[V_{C'}]$ inherit its
 embedding from $\Pi_G$.
 
-Then $C$, with the inherited embedding, is a tire graph in the sense of Definition~\ref{def:tire-graph}.  Its outer boundary
+Then $T_{C'}$, with the inherited embedding, is a tire graph in the sense of Definition~\ref{def:tire-graph}.  Its outer boundary
 $B_{\mathrm{out}}$ is the side of $R_{C'}$ closer to $S$ in $\Pi_G$,
 namely the level-$d$ subgraph $G[V_{C'} \cap L_d]$ (a simple cycle or
 single vertex); its inner outerplanar graph is $O = G[V_{C'} \cap
 L_{d+1}]$, and its inner boundary $B_{\mathrm{in}}$ is the outer-face
 boundary closed walk of $O$ in the inherited embedding (a simple cycle
 when $O$ is $2$-connected, a non-simple closed walk in general).  The
-triangular faces of $C$ inside the closed boundary region are exactly
+triangular faces of $T_{C'}$ inside the closed boundary region are exactly
 the faces of $G$ in $F_{C'}$.
 \end{lemma}
 
@@ -325,7 +325,7 @@ both outerplanar.
 at level $d$, and adjacent vertices in $G$ differ in level by at most
 $1$; combined with $\delta_G(d_f) = d$, this forces
 $V(f) \subseteq L_d \cup L_{d+1}$.  Hence $V_{C'} \subseteq L_d \cup
-L_{d+1}$, and $C$ has vertex partition
+L_{d+1}$, and $T_{C'}$ has vertex partition
 $V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap L_{d+1})$.
 
 \emph{Boundary edges are monochromatic in level.}  Each edge $e$ on
@@ -370,7 +370,7 @@ necessarily a simple cycle).
 
 \emph{Outer boundary.}  Because $S$ lies on the outer face of $\Pi_G$,
 the boundary curve(s) of $R_{C'}$ on the $L_d$ side are closer to $S$
-in the embedding.  In the inherited embedding of $C$, the unique
+in the embedding.  In the inherited embedding of $T_{C'}$, the unique
 unbounded face is the merged region containing the rest of $\Pi_G$
 outside $R_{C'}$ on the $S$ side, so its boundary --- a simple cycle
 on $L_d$ (or a single vertex when $V_{C'} \cap L_d = \{v_0\}$, the
@@ -393,9 +393,9 @@ precisely to the multiple connected components or bridge crossings of
 $O$, and the outer-face boundary closed walk of $O$ captures them
 collectively.
 
-\emph{Tire structure.}  The triangular faces of $C$ inside the closed
+\emph{Tire structure.}  The triangular faces of $T_{C'}$ inside the closed
 boundary region are by construction the depth-$d$ faces in $F_{C'}$,
-and the edges of $C$ are $E(B_{\mathrm{out}}) \cup E(O) \cup
+and the edges of $T_{C'}$ are $E(B_{\mathrm{out}}) \cup E(O) \cup
 E_{\mathrm{ann}}$ where $E_{\mathrm{ann}}$ are the edges of $G$
 between $V_{C'} \cap L_d$ and $V_{C'} \cap L_{d+1}$ that bound a face
 of $F_{C'}$.
@@ -456,7 +456,7 @@ where $C'$ is its component of $G'_d$.  So $\bigcup_{R \in
 
 \begin{remark}
 \label{rem:tire-component-degenerate}
-Either boundary part of $C$ in Lemma~\ref{lem:tire-component} may be
+Either boundary part of $T_{C'}$ in Lemma~\ref{lem:tire-component} may be
 degenerate.  At $d = 0$ with single-vertex source $S = \{v_0\}$ the
 unique component of $G'_0$ has $V_{C'} \cap L_0 = \{v_0\}$ as the
 degenerate \emph{outer} boundary and $V_{C'} \cap L_1$ a cycle (the