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papers: name R the "tire tread" in Definition 1.5

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Foundational paper: Definition 1.5 (Tire graph) now explicitly
names the closed planar region R bounded by B_out and B_in the
"tire tread of T". Remark 1.6 and the abstract updated to use
the new term.

Dual paper: places that referred to R as "the closed annular
region" or "the annular region" updated to use "tire tread" for
consistency:
  - Definition 1.1 (Partial tire dual)
  - Caption of figure on partial tire dual example
  - Two places inside the proof of Proposition 1.2

"annular edges" (E_ann) and "annular faces" (F_ann) kept as-is
since they're established notation; the tread is the region they
triangulate.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 01:01:32 -04:00
parent 70b605e84f
commit 9b4100e143
7 changed files with 30 additions and 29 deletions
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papers/coloring_nested_tire_dual_graphs/paper.aux
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\citation{bauerfeld-nested-tires}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\newlabel{def:partial-tire-dual}{{1.1}{1}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{2}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The partial tire dual $D(T)$ (purple squares + orange diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$ and $k = 4$. The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the tire tread (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.2\hbox {}.}}{2}{}\protected@file@percent }
\newlabel{fig:partial-tire-dual-example}{{1}{2}}
\newlabel{prop:partial-tire-dual-structure}{{1.2}{2}}
\citation{bauerfeld-nested-tires}
papers/coloring_nested_tire_dual_graphs/paper.log
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papers/coloring_nested_tire_dual_graphs/paper.tex
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@@ -100,8 +100,8 @@ and $G$ has $2n - 4$ triangular faces.
Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in
the sense of \cite[Definition~1.5]{bauerfeld-nested-tires}, 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
denote the set of triangular faces of $T$ in the tire tread
(the closed 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.}
@@ -145,7 +145,7 @@ tire dual} of $T$, written $D(T)$, is the graph defined as follows.
diamonds) drawn on top of a small tire graph $T$ (faint) with $m = 6$
and $k = 4$. The ten interior vertices $d_f$ at the centroids of the
annular triangles form a single $10$-cycle (solid purple); each
boundary edge of the annular region (either of $B_{\mathrm{out}}$ or
boundary edge of the tire tread (either of $B_{\mathrm{out}}$ or
of $B_{\mathrm{in}}$) contributes a degree-$1$ leaf (orange diamond)
attached to the unique annular face incident to it (dashed orange),
giving the structure $C_{10} \circ K_1$ of
@@ -198,13 +198,13 @@ $\{x, y, z\}$ 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
$B_{\mathrm{in}}$) is a boundary edge of the tire tread; 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
connectedness of the tire tread this forces it to be a single
cycle. By \cite[Remark~1.6]{bauerfeld-nested-tires}, the cycle has length $n + m$,
and there are also $n + m$ leaves attached one-per-cycle-vertex.
\end{proof}
papers/coloring_nested_tire_graphs/paper.log
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -50,9 +50,9 @@ triangulations. A \emph{level source} of a triangulation $G$
induces a BFS layering of $G$, which in turn endows the inner
planar dual $G'$ with a \emph{dual depth} grading. We isolate the
basic object of study --- the \emph{tire graph} $T$, a plane graph
whose outer and inner boundaries bound an annular region
triangulated by the \emph{annular edges} $E_{\mathrm{ann}}$ --- and
record its face/edge counts.
whose outer and inner boundaries bound the \emph{tire tread} $R$,
a closed region triangulated by the \emph{annular edges}
$E_{\mathrm{ann}}$ --- and record its face/edge counts.
\end{abstract}
\maketitle
@@ -150,20 +150,21 @@ The vertex and edge sets of $T$ are
E(T) = E(B_{\mathrm{out}}) \cup E(O) \cup E_{\mathrm{ann}},
\]
where $E_{\mathrm{ann}}$ --- the \emph{annular edges} --- has the
property that, in the plane embedding of $T$, the closed planar region
$R$ bounded externally by $B_{\mathrm{out}}$ and internally by
$B_{\mathrm{in}}$ is partitioned into triangular faces of $T$ whose
union is $R$.
property that, in the plane embedding of $T$, the closed planar
region $R$ bounded externally by $B_{\mathrm{out}}$ and internally
by $B_{\mathrm{in}}$ is partitioned into triangular faces of $T$
whose union is $R$. We call $R$ the \emph{tire tread} of $T$.
When $B_{\mathrm{out}}$ is a simple cycle and $O$ is $2$-connected,
$R$ is a closed annulus. More generally, $R$ is a closed planar
region that may fail to be a $2$-manifold at cut-vertices of $O$ (where
two ``lobes'' of the depth-$d$ region meet at a single vertex); the
inner boundary $B_{\mathrm{in}}$ is then a non-simple closed walk that
visits the cut-vertex multiple times. The relaxed definition
accommodates outerplanar inner graphs with bridges, cut-vertices, or
multiple connected components. When either boundary is degenerate,
$R$ is a closed disk with that vertex as apex.
the tread is a closed annulus. More generally, $R$ is a closed
planar region that may fail to be a $2$-manifold at cut-vertices of
$O$ (where two ``lobes'' of the depth-$d$ region meet at a single
vertex); the inner boundary $B_{\mathrm{in}}$ is then a non-simple
closed walk that visits the cut-vertex multiple times. The relaxed
definition accommodates outerplanar inner graphs with bridges,
cut-vertices, or multiple connected components. When either
boundary is degenerate, the tread is a closed disk with that vertex
as apex.
\end{definition}
\begin{figure}[h]
@@ -182,8 +183,8 @@ triangular faces.}
\begin{remark}
\label{rem:tire-counts}
Let $m = |V(B_{\mathrm{out}})|$ and $k = |V(B_{\mathrm{in}})|$. By
Euler's formula on the annular (resp.\ disk) region $R$, the tire graph
has $m + k$ triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$
Euler's formula on the tire tread $R$, the tire graph has $m + k$
triangular faces inside $R$ and $|E_{\mathrm{ann}}| = m + k$
annular edges when neither boundary is degenerate; when exactly one
boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.