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

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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>
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didericis
2026-06-01 00:18:04 -04:00
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commit 2bbd863643
3 changed files with 11 additions and 11 deletions
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papers/coloring_nested_tire_graphs/paper.log
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papers/coloring_nested_tire_graphs/paper.pdf
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -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 single-vertex source). For $d \geq 0$, let $C'$ be a connected
component of the depth-$d$ dual subgraph $G'_d$, with faces $F_{C'}$, component of the depth-$d$ dual subgraph $G'_d$, with faces $F_{C'}$,
bounding vertices $V_{C'}$, and region $R_{C'}$ as in 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 $\Pi_G$. 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 $\Pi_G$, $B_{\mathrm{out}}$ 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 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 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 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 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 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 $F_{C'}$. the faces of $G$ in $F_{C'}$.
\end{lemma} \end{lemma}
@@ -325,7 +325,7 @@ both outerplanar.
at level $d$, and adjacent vertices in $G$ differ in level by at most at level $d$, and adjacent vertices in $G$ differ in level by at most
$1$; combined with $\delta_G(d_f) = d$, this forces $1$; 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 $V(f) \subseteq L_d \cup L_{d+1}$. Hence $V_{C'} \subseteq L_d \cup
L_{d+1}$, and $C$ has vertex partition L_{d+1}$, and $T_{C'}$ has vertex partition
$V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap L_{d+1})$. $V_{C'} = (V_{C'} \cap L_d) \sqcup (V_{C'} \cap L_{d+1})$.
\emph{Boundary edges are monochromatic in level.} Each edge $e$ on \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 $\Pi_G$, \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 $S$ 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 $\Pi_G$ 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 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 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 $O$, and the outer-face boundary closed walk of $O$ captures them
collectively. 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 $F_{C'}$, 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 $G$ 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 between $V_{C'} \cap L_d$ and $V_{C'} \cap L_{d+1}$ that bound a face
of $F_{C'}$. of $F_{C'}$.
@@ -456,7 +456,7 @@ where $C'$ is its component of $G'_d$. So $\bigcup_{R \in
\begin{remark} \begin{remark}
\label{rem:tire-component-degenerate} \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 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 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 degenerate \emph{outer} boundary and $V_{C'} \cap L_1$ a cycle (the