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coloring_nested_tire_graphs: actually apply the title + tire definition edits

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The previous rename commit (6ca0c6d) staged the unmodified paper.tex
content because `git mv` + `git add` picked up the on-disk file as it
was at HEAD, not the unstaged working-tree edits.  This commit applies
what 6ca0c6d's message claimed:

- Title: "Nested Level Duals" → "Coloring Nested Tire Graphs"
- Adds Definition 1.5 (Tire graph) formalising (C_out, O, E_ann) with
  the annular-triangulation condition, plus a Remark on vertex/edge/
  face counts.
- Removes the 2026-05-22 "shelved" note.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-25 14:32:12 -04:00
parent 6ca0c6dd15
commit c0e71e2d25
5 changed files with 68 additions and 34 deletions
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papers/coloring_nested_tire_graphs/paper.aux
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\newlabel{def:dual-depth}{{1.4}{1}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Dual depth in a stacked-ring triangulation $G$ with level source $S = \{0\}$. Each $G$ vertex is labelled by its level $\ell $. Each bounded face carries a dual vertex (square, joined by dashed dual edges) coloured by its dual depth $\delta (d_f) = \qopname \relax m{min}_{v \in V(f)} \ell (v)$: the central fan has depth $0$, the inner annulus depth $1$, and the outer annulus depth $2$. The outer face (the level-$3$ triangle) is excluded from the inner dual and carries no dual vertex.}}{2}{}\protected@file@percent }
\newlabel{fig:dual-depth}{{1}{2}}
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\newlabel{def:tire-graph}{{1.5}{2}}
\gdef \@abspage@last{3}
papers/coloring_nested_tire_graphs/paper.fdb_latexmk
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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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\begin{document}
\title{Nested Level Duals}
\title{Coloring Nested Tire Graphs}
% author one information
\author{Eric Bauerfeld}
@@ -114,8 +114,41 @@ vertex.}
\label{fig:dual-depth}
\end{figure}
\end{document}
\begin{definition}[Tire graph]
\label{def:tire-graph}
Let $C_{\mathrm{out}}$ be a simple cycle of length $m \geq 3$, and let
$O$ be an outerplanar graph whose outer-face boundary $C_{\mathrm{in}}$
is a simple cycle of length $k \geq 3$, with $V(C_{\mathrm{out}}) \cap
V(O) = \emptyset$. A \emph{tire graph} on $(C_{\mathrm{out}}, O)$ is a
plane graph $T$ with
\[
V(T) = V(C_{\mathrm{out}}) \cup V(O),
\qquad
E(T) = E(C_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}},
\]
where $E_{\mathrm{ann}}$ is a set of edges --- the \emph{annular edges}
--- such that, in the plane embedding of $T$, the closed annulus with
outer boundary $C_{\mathrm{out}}$ and inner boundary $C_{\mathrm{in}}$
is partitioned into triangular faces. Equivalently, the bounded faces
of $T$ that are not faces of $O$ are all triangles, and together they
tile the annular region between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$.
% NOTE (2026-05-22): This paper is being shelved in favour of an alternative
% approach. The nested-level-duals framing is preserved here for reference but
% is not being actively developed.
We call $C_{\mathrm{out}}$ the \emph{outer cycle}, $O$ the \emph{inner
outerplanar graph}, and $C_{\mathrm{in}}$ the \emph{inner cycle} of
$T$. When $O = C_{\mathrm{in}}$ (the inner outerplanar graph has no
chords), $T$ is a tire graph \emph{with empty inner}; in general $O$
contributes only chords inside the disk bounded by $C_{\mathrm{in}}$
and does not interact with $E_{\mathrm{ann}}$.
\end{definition}
\begin{remark}
A tire graph on $(C_{\mathrm{out}}, O)$ has $|V(C_{\mathrm{out}})| +
|V(O)| = m + k$ vertices, exactly $m + k$ annular triangles
in the annulus between $C_{\mathrm{out}}$ and $C_{\mathrm{in}}$ (by
Euler's formula on the annulus), and exactly $m + k$ annular edges
in $E_{\mathrm{ann}}$, of which the $m + k$ triangles share their
three edges with the boundaries $E(C_{\mathrm{out}}) \cup
E(C_{\mathrm{in}})$ and with each other.
\end{remark}
\end{document}