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coloring_nested_tire_graphs: add Definition 1.7 (Partial tire dual) + structure proposition

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Adds Definition 1.7 (Partial tire dual) formalising the user's
construction: for a tire graph T with annular face set F_{ann}, the
partial tire dual D(T) has

  - Interior vertices d_f for each annular face f,
  - Leaf vertices for each edge of B_out and each occurrence of an
    edge on the boundary walk B_in (so cut-vertices/bridges of O
    contribute multiple leaves),
  - Interior dual edges for each annular edge incident to two annular
    faces,
  - Leaf edges from d_f to the corresponding leaf for each boundary
    edge of the annular region.

Adds Proposition 1.8 showing that when the annular triangulation is
spoke-only (i.e. every annular edge has one endpoint on B_out and one
on B_in) and O is 2-connected, each annular face has exactly 1
boundary edge + 2 interior annular edges.  Consequently each interior
vertex d_f has degree 3 = 2 (cycle) + 1 (leaf), and the induced
subgraph on {d_f} is a single cycle of length n + m.  D(T) is then
isomorphic to the corona C_{n+m} ∘ K_1 -- a cycle of length n+m with
one leaf attached to each cycle vertex; |V(D(T))| = |E(D(T))| = 2(n+m).

Subsequent numbering shifted: Proposition (Source-side simple-cycle
property) is now 1.9; Lemma (Tire-component) is now 1.10; Remarks
shift to 1.11 and 1.12.  All cross-references are by label, so they
update automatically.

Paper grows from 6 to 7 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-25 18:00:05 -04:00
parent c2961b1833
commit e4216ec7a2
4 changed files with 111 additions and 36 deletions
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papers/coloring_nested_tire_graphs/paper.aux
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A tire graph with non-degenerate boundaries: outer boundary $B_{\mathrm {out}}$ a $6$-cycle on vertices $0,\dots ,5$ (blue), inner boundary $B_{\mathrm {in}}$ a $4$-cycle on vertices $6,\dots ,9$ (red), inner outerplanar graph $O = B_{\mathrm {in}} \cup \{7\text {--}9\}$ (with one chord, orange), and $E_{\mathrm {ann}}$ (grey) tiling the annulus between $B_{\mathrm {out}}$ and $B_{\mathrm {in}}$ by ten triangular faces.}}{3}{}\protected@file@percent }
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\newlabel{rem:tire-counts}{{1.6}{3}}
\newlabel{prop:no-level-d-pinch}{{1.7}{3}}
\newlabel{def:partial-tire-dual}{{1.7}{3}}
\newlabel{prop:partial-tire-dual-structure}{{1.8}{4}}
\newlabel{prop:no-level-d-pinch}{{1.9}{4}}
\citation{bauerfeld-pds}
\newlabel{lem:tire-component}{{1.8}{4}}
\newlabel{lem:tire-component}{{1.10}{5}}
\citation{bauerfeld-pds}
\newlabel{rem:tire-component-degenerate}{{1.9}{5}}
\newlabel{rem:tire-component-degenerate}{{1.11}{6}}
\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{6}}
\bibcite{bauerfeld-pds}{1}
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papers/coloring_nested_tire_graphs/paper.log
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@@ -178,6 +178,82 @@ 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{proposition}[Structure of $D(T)$ when the annular triangulation
is spoke-only]
\label{prop:partial-tire-dual-structure}
Suppose $B_{\mathrm{out}}$ is a simple cycle of length $n$, $O$ is a
$2$-connected outerplanar graph whose outer-face cycle
$B_{\mathrm{in}}$ has length $m$, and $E_{\mathrm{ann}}$ consists only
of \emph{spokes} (edges with one endpoint in $V(B_{\mathrm{out}})$ and
one in $V(B_{\mathrm{in}})$). Then each face $f \in F_{\mathrm{ann}}$
has exactly one boundary edge (on $B_{\mathrm{out}}$ or
$B_{\mathrm{in}}$) and two interior annular edges, and consequently
$D(T)$ is isomorphic to the corona graph $C_{n+m} \circ K_1$: a cycle
of length $n + m$ on the interior vertices $\{d_f\}$, with one leaf
attached to each cycle vertex.
In particular, $|V(D(T))| = 2(n+m)$ and $|E(D(T))| = 2(n+m)$.
\end{proposition}
\begin{proof}
Each annular triangle $f$ in a spoke-only triangulation has the form
$\{x, y, z\}$ 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
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
cycle. By Remark~\ref{rem:tire-counts}, the cycle has length $n + m$,
and there are also $n + m$ leaves attached one-per-cycle-vertex.
\end{proof}
\begin{proposition}[Source-side simple-cycle property]
\label{prop:no-level-d-pinch}
Let $G$ be a maximal planar graph with planar embedding $\Pi_G$ and