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papers: move Definition 1.7 (partial tire dual) to dual paper

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REMOVED from coloring_nested_tire_graphs/:
  - Definition 1.7 (Partial tire dual)
  - Figure 3 (partial tire dual example)
  - Figure 4 (partial tire dual bridge case)
  - fig_partial_tire_dual.png file
  - fig_partial_tire_dual_bridge.png file
  - Abstract no longer mentions partial tire dual

Foundational paper now ends at Remark 1.6 (tire face/edge counts).
Down from 5 to 3 pages.

ADDED to coloring_nested_tire_dual_graphs/:
  - Definition (Partial tire dual) — now numbered 1.1 in this paper
  - Figure: partial tire dual example
  - Figure: partial tire dual bridge case
  - Both PNG figure files

Inserted before the structure proposition (former 1.1, now 1.2).
Intro citation list removes the bullet for partial tire dual since
it's now defined locally. The definition's internal ref to
Definition~\ref{def:tire-graph} becomes
\cite[Definition~1.5]{bauerfeld-nested-tires}.

The two figure captions updated to reference
prop:partial-tire-dual-structure locally (instead of citing the
companion paper as if it owned the definition).

Paper grows from 8 to 9 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-27 00:57:31 -04:00
parent 65f79f2e65
commit df9411d2bd
10 changed files with 195 additions and 203 deletions
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papers/coloring_nested_tire_dual_graphs/paper.aux
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@@ -7,31 +7,34 @@
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
\newlabel{prop:partial-tire-dual-structure}{{1.1}{1}}
\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 }
\newlabel{fig:partial-tire-dual-example}{{1}{2}}
\newlabel{prop:partial-tire-dual-structure}{{1.2}{2}}
\citation{bauerfeld-nested-tires}
\newlabel{prop:no-level-d-pinch}{{1.2}{2}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.2\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{3}{}\protected@file@percent }
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\citation{bauerfeld-nested-tires}
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\citation{bauerfeld-nested-tires}
\citation{bauerfeld-depth}
\citation{bauerfeld-nested-tires}
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\citation{bauerfeld-depth}
\citation{bauerfeld-nested-tires}
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\newlabel{rem:tire-no-extra-hypotheses}{{1.5}{4}}
\newlabel{prop:edge-vertex-bijection}{{1.6}{4}}
\newlabel{rem:tire-component-degenerate}{{1.5}{6}}
\newlabel{rem:tire-no-extra-hypotheses}{{1.6}{6}}
\newlabel{prop:edge-vertex-bijection}{{1.7}{6}}
\citation{bauerfeld-nested-tires}
\citation{bauerfeld-nested-tires}
\newlabel{rem:edge-vertex-corollary}{{1.7}{5}}
\newlabel{def:tire-annular-subgraph}{{1.8}{5}}
\newlabel{def:tire-annular-face-connector}{{1.9}{5}}
\newlabel{def:spokes}{{1.10}{6}}
\newlabel{rem:facial-dual-spoke-only}{{1.11}{6}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{7}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The bridge case: $T'_{\mathrm {ann}} = \theta (1, 3, 3)$ has three faces $A, B, C$ in its inherited embedding, with respective vertex sets $V(A) = \{v_0, \dots , v_5\}$, $V(B) = \{v_0, v_1, v_2, v_3\}$, and $V(C) = \{v_0, v_3, v_4, v_5\}$. In the surrounding maximal planar $G$, the chord endpoints $v_0, v_3$ (the two annular faces sharing the bridge edge) have all three $G'$-edges inside $T'_{\mathrm {ann}}$, while each non-chord vertex $v_i$ ($i \in \{1, 2, 4, 5\}$) contributes one $G'$-edge to an external non-annular neighbor $u_i$. Each panel highlights $T'_{f'}$ (blue) inside $G'$: dark circles are $V(f')$, gray circles are $G'$-neighbors of $V(f')$ within $T'_{\mathrm {ann}}$, and red squares are external $G'$-neighbors $u_i$. The choice of face $f'$ controls which external neighbors $u_i$ are pulled into $T'_{f'}$ (face $A$ pulls in all four; face $B$ pulls in $u_1, u_2$ and face $C$ pulls in $u_4, u_5$).}}{8}{}\protected@file@percent }
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\bibcite{bauerfeld-depth}{1}
\bibcite{bauerfeld-nested-tires}{2}
\newlabel{tocindent-1}{0pt}
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@@ -86,8 +86,6 @@ use, without restating, the notions of:
\item \emph{tire graph} $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$
with outer/inner boundaries and annular edges
(\cite[Definition~1.5]{bauerfeld-nested-tires});
\item \emph{partial tire dual} $D(T)$
(\cite[Definition~1.7]{bauerfeld-nested-tires});
\item face/edge counts
(\cite[Remark~1.6]{bauerfeld-nested-tires}).
\end{itemize}
@@ -97,6 +95,86 @@ with a fixed planar embedding $\Pi_G$. We write $|V| = n$, so $|E| = 3n - 6$
and $G$ has $2n - 4$ triangular faces.
\begin{definition}[Partial tire dual]
\label{def:partial-tire-dual}
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
tire dual} of $T$, written $D(T)$, is the graph defined as follows.
\emph{Vertices.}
\begin{enumerate}
\item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an
\emph{interior vertex} $d_f$ of $D(T)$.
\item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a
\emph{leaf vertex} $\ell_e^{\mathrm{out}}$.
\item[(V3)] For each occurrence of an edge in the closed walk
$B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$),
a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$. (When $O$ is
$2$-connected each edge appears once; cut-vertices and
bridges of $O$ may cause an edge or vertex to appear more
than once.)
\end{enumerate}
\emph{Edges.}
\begin{enumerate}
\item[(E1)] For each edge $e \in E(T)$ whose two incident faces both
lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}),
one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where
$f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces
incident to $e$.
\item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge
$\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where
$f \in F_{\mathrm{ann}}$ is the unique annular face incident
to $e$. The leaf $\ell_e^{\mathrm{out}}$ has degree $1$.
\item[(E3)] For each occurrence of $e$ on the boundary walk
$B_{\mathrm{in}}$, one edge
$\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where
$f \in F_{\mathrm{ann}}$ is the annular face incident to $e$
on the side of that occurrence. The leaf
$\ell_e^{\mathrm{in}}$ has degree $1$.
\end{enumerate}
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual.png}
\caption{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~\ref{prop:partial-tire-dual-structure}.}
\label{fig:partial-tire-dual-example}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png}
\caption{Partial tire dual $D(T)$ when the inner outerplanar graph
$O$ has a bridge --- here a non-trivial edge cut connecting two
disjoint triangles. $B_{\mathrm{out}}$ is a $4$-cycle on
$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together
with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing
the bridge disconnects $O$). Because both faces incident to the
bridge are annular triangles, the bridge contributes an
\emph{interior dual edge} (highlighted in red) rather than two
leaves; consequently the interior dual subgraph is no longer the
single $(n+m)$-cycle of
Proposition~\ref{prop:partial-tire-dual-structure}, but a theta
graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident
annular faces) are joined by three internally vertex-disjoint paths
in $D(T)$. Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves)
and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves,
three for each triangle).}
\label{fig:partial-tire-dual-bridge}
\end{figure}
\begin{proposition}[Structure of $D(T)$ when the annular triangulation
is spoke-only]
\label{prop:partial-tire-dual-structure}
papers/coloring_nested_tire_graphs/fig_partial_tire_dual.png
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papers/coloring_nested_tire_graphs/paper.tex
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@@ -52,9 +52,7 @@ 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
define its \emph{partial tire dual} $D(T)$, the dual restricted to
$T$'s annular faces together with leaves recording the boundary
edges.
record its face/edge counts.
\end{abstract}
\maketitle
@@ -191,83 +189,6 @@ boundary is degenerate (so $\min(m, k) = 1$), there are $m + k - 1$
triangular faces and $|E_{\mathrm{ann}}| = m + k - 1$.
\end{remark}
\begin{definition}[Partial tire dual]
\label{def:partial-tire-dual}
Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph in
the sense of Definition~\ref{def:tire-graph}, 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
tire dual} of $T$, written $D(T)$, is the graph defined as follows.
\emph{Vertices.}
\begin{enumerate}
\item[(V1)] For each face $f \in F_{\mathrm{ann}}$, an
\emph{interior vertex} $d_f$ of $D(T)$.
\item[(V2)] For each edge $e \in E(B_{\mathrm{out}})$, a
\emph{leaf vertex} $\ell_e^{\mathrm{out}}$.
\item[(V3)] For each occurrence of an edge in the closed walk
$B_{\mathrm{in}}$ (= the outer-face boundary walk of $O$),
a \emph{leaf vertex} $\ell_e^{\mathrm{in}}$. (When $O$ is
$2$-connected each edge appears once; cut-vertices and
bridges of $O$ may cause an edge or vertex to appear more
than once.)
\end{enumerate}
\emph{Edges.}
\begin{enumerate}
\item[(E1)] For each edge $e \in E(T)$ whose two incident faces both
lie in $F_{\mathrm{ann}}$ (an \emph{interior annular edge}),
one edge $\{d_{f_1}, d_{f_2}\} \in E(D(T))$ where
$f_1, f_2 \in F_{\mathrm{ann}}$ are the two annular faces
incident to $e$.
\item[(E2)] For each $e \in E(B_{\mathrm{out}})$, one edge
$\{d_f, \ell_e^{\mathrm{out}}\} \in E(D(T))$ where
$f \in F_{\mathrm{ann}}$ is the unique annular face incident
to $e$. The leaf $\ell_e^{\mathrm{out}}$ has degree $1$.
\item[(E3)] For each occurrence of $e$ on the boundary walk
$B_{\mathrm{in}}$, one edge
$\{d_f, \ell_e^{\mathrm{in}}\} \in E(D(T))$ where
$f \in F_{\mathrm{ann}}$ is the annular face incident to $e$
on the side of that occurrence. The leaf
$\ell_e^{\mathrm{in}}$ has degree $1$.
\end{enumerate}
\end{definition}
\begin{figure}[h]
\centering
\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual.png}
\caption{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 $C_{10} \circ K_1$ analysed in the companion
paper~\cite{bauerfeld-nested-tire-duals}.}
\label{fig:partial-tire-dual-example}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png}
\caption{Partial tire dual $D(T)$ when the inner outerplanar graph
$O$ has a bridge --- here a non-trivial edge cut connecting two
disjoint triangles. $B_{\mathrm{out}}$ is a $4$-cycle on
$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together
with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing
the bridge disconnects $O$). Because both faces incident to the
bridge are annular triangles, the bridge contributes an
\emph{interior dual edge} (highlighted in red) rather than two
leaves; consequently the interior dual subgraph is no longer the
single $(n+m)$-cycle of the spoke-only case, but a theta
graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident
annular faces) are joined by three internally vertex-disjoint paths
in $D(T)$. Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves)
and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves,
three for each triangle).}
\label{fig:partial-tire-dual-bridge}
\end{figure}
\begin{thebibliography}{9}