coloring_nested_tire_graphs: figures for Theorem 1.12 (Case 1 + Case 2)
Two TikZ figures added to the outerplanarity theorem: Figure (Case 1, disk tread): apex v_0 at center, hexagonal non-degenerate boundary (red), 6 spokes (grey) forming a fan of 6 triangles. Dual Γ (blue) is the cycle C_6 connecting the 6 triangle centroids. Outerplanar trivially. Figure (Case 2, annulus tread): two concentric hexagons for B_out and B_in, spokes + one extra "bridge-style" interior annular edge. Dual Γ is a Hamilton cycle of length 12 around the annulus, plus one chord (dashed). All vertices on outer face → outerplanar. Also corrected the Case 1 proof: the disk has a single interior vertex (the apex), so the triangulation is a FAN around the apex (not a polygon-triangulation with no interior vertices), and Γ is a cycle of length k (not a tree). This is still outerplanar. Added tikz + backgrounds library to preamble. Page count: 8 → 9. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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\usepackage{amssymb}
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\usepackage{graphicx}
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\usepackage{tikz}
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\usetikzlibrary{backgrounds}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{lemma}[theorem]{Lemma}
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@@ -487,14 +489,67 @@ annulus.
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\medskip
|
||||
\emph{Case 1: $R$ is a closed disk} (one of $B_{\mathrm{out}},
|
||||
B_{\mathrm{in}}$ degenerate, by Definition~\ref{def:tire-graph}).
|
||||
Then $T \cap R$ triangulates a polygon with no interior vertices:
|
||||
the polygon's boundary cycles round the unique non-degenerate
|
||||
boundary plus the degenerate apex, and all vertices of $V(T)
|
||||
\cap R$ lie on this polygon. The dual graph of such a polygon
|
||||
triangulation is a tree (a classical fact: a triangulation of a
|
||||
$p$-gon with no interior vertex has $p - 2$ triangles and $p - 3$
|
||||
diagonals, and the diagonals' adjacency graph is a tree). Trees
|
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are outerplanar.
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||||
Let $v_0$ be the degenerate-boundary vertex (the apex) and let
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||||
$k = |B_{\mathrm{non-deg}}|$ be the length of the non-degenerate
|
||||
boundary cycle. The triangulation of $R$ is a \emph{fan} of $k$
|
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triangles around $v_0$: each triangle has the form $\{v_0, u_i,
|
||||
u_{i+1}\}$ where $u_1, \dots, u_k$ are the boundary-cycle vertices
|
||||
in cyclic order. Each triangle has two spoke edges (= the two
|
||||
edges incident to $v_0$, shared with the two neighbouring fan
|
||||
triangles) and one boundary edge (in $B_{\mathrm{non-deg}}$,
|
||||
contributing a leaf in $D(T)$ but no edge in $\Gamma$). Hence
|
||||
every $d_f$ has $\Gamma$-degree exactly $2$, and $\Gamma$ is a
|
||||
single cycle of length $k$. Cycles are outerplanar.
|
||||
|
||||
See Figure~\ref{fig:inner-dual-disk-case} for the disk case
|
||||
($k = 6$).
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{tikzpicture}[scale=1.4]
|
||||
\def\R{1.8}
|
||||
% apex
|
||||
\node[circle, fill=black, inner sep=1.6pt, label={right:$v_0$}] (apex) at (0, 0) {};
|
||||
% boundary vertices (hexagon)
|
||||
\foreach \i in {0,...,5} {
|
||||
\pgfmathsetmacro{\ang}{60*\i + 90}
|
||||
\node[circle, fill=black, inner sep=1.3pt] (u\i) at (\ang:\R) {};
|
||||
}
|
||||
% boundary cycle edges (non-degenerate boundary)
|
||||
\foreach \i in {0,...,5} {
|
||||
\pgfmathtruncatemacro{\j}{mod(\i+1,6)}
|
||||
\draw[red, thick] (u\i) -- (u\j);
|
||||
}
|
||||
% spoke edges (annular interior)
|
||||
\foreach \i in {0,...,5} {
|
||||
\draw[gray] (apex) -- (u\i);
|
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}
|
||||
% dual vertices at triangle centroids + dual cycle
|
||||
\foreach \i in {0,...,5} {
|
||||
\pgfmathsetmacro{\angmid}{60*\i + 90 + 30}
|
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\pgfmathsetmacro{\rmid}{0.62*\R}
|
||||
\node[circle, fill=blue!70!black, inner sep=1.6pt] (d\i) at (\angmid:\rmid) {};
|
||||
}
|
||||
\foreach \i in {0,...,5} {
|
||||
\pgfmathtruncatemacro{\j}{mod(\i+1,6)}
|
||||
\draw[blue!70!black, very thick] (d\i) -- (d\j);
|
||||
}
|
||||
% Labels
|
||||
\node[red] at (0, -\R - 0.3) {\small non-degenerate boundary $B_{\mathrm{non-deg}}$};
|
||||
\node[blue!70!black] at (\R + 1.0, 0.5) {\small dual cycle $\Gamma \cong C_6$};
|
||||
\node[blue!70!black] at (\R + 1.0, 0.2) {\small (outerplanar)};
|
||||
\node[gray] at (-\R - 0.4, 0.3) {\small spokes};
|
||||
\end{tikzpicture}
|
||||
\caption{Case 1 ($R$ = disk, $k = 6$). The apex $v_0$ sits at the
|
||||
centre; the non-degenerate boundary $B_{\mathrm{non-deg}}$ (red)
|
||||
is the hexagonal outer cycle; spokes (grey) triangulate the disk
|
||||
into a fan of $6$ triangles around $v_0$. Each triangle has two
|
||||
spoke edges (interior, contributing $\Gamma$-edges) and one
|
||||
boundary edge (contributing a leaf in $D(T)$, no $\Gamma$-edge).
|
||||
The inner dual $\Gamma$ (blue) is the cycle $C_6$ formed by the
|
||||
six annular face centroids, a manifestly outerplanar graph.}
|
||||
\label{fig:inner-dual-disk-case}
|
||||
\end{figure}
|
||||
|
||||
\medskip
|
||||
\emph{Case 2: $R$ is an annulus} (both $B_{\mathrm{out}}$ and
|
||||
@@ -580,6 +635,71 @@ on the outer face (the unbounded region outside the circle),
|
||||
making $\Gamma$ outerplanar. $\square$
|
||||
\end{proof}
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\begin{tikzpicture}[scale=1.35]
|
||||
\def\Rout{2.0}
|
||||
\def\Rin{1.05}
|
||||
% Boundary cycles
|
||||
\foreach \i in {0,...,5} {
|
||||
\pgfmathsetmacro{\ang}{60*\i + 90}
|
||||
\node[circle, fill=black, inner sep=1.2pt] (uo\i) at (\ang:\Rout) {};
|
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}
|
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\foreach \i in {0,...,5} {
|
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|
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|
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}
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|
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|
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\draw[red, thick] (uo\i) -- (uo\j);
|
||||
\draw[red!60!white, thick] (ui\i) -- (ui\j);
|
||||
}
|
||||
% Annular edges: spokes (each outer vertex connects to 2 inner)
|
||||
\foreach \i in {0,...,5} {
|
||||
\pgfmathtruncatemacro{\j}{mod(\i,6)}
|
||||
\pgfmathtruncatemacro{\k}{mod(\i+5,6)}
|
||||
\draw[gray] (uo\i) -- (ui\j);
|
||||
\draw[gray] (uo\i) -- (ui\k);
|
||||
}
|
||||
% Highlight one bridge-like annular interior edge — between two inner vertices
|
||||
% (For illustration we use the "bridge" between inner i=0 and i=3)
|
||||
\draw[gray, dashed, thick] (ui0) to[bend right=15] (ui3);
|
||||
% Dual: 12 annular triangles → 12 dual vertices arranged between
|
||||
\foreach \i in {0,...,5} {
|
||||
\pgfmathsetmacro{\ango}{60*\i + 90 - 15}
|
||||
\pgfmathsetmacro{\rmido}{0.5*\Rout + 0.5*\Rin}
|
||||
\node[circle, fill=blue!70!black, inner sep=1.4pt] (do\i) at (\ango:\rmido) {};
|
||||
\pgfmathsetmacro{\angi}{60*\i + 90 + 15}
|
||||
\node[circle, fill=blue!70!black, inner sep=1.4pt] (di\i) at (\angi:\rmido) {};
|
||||
}
|
||||
% Dual cycle: do0 - di0 - do1 - di1 - ... around
|
||||
\foreach \i in {0,...,5} {
|
||||
\pgfmathtruncatemacro{\j}{mod(\i+1,6)}
|
||||
\draw[blue!70!black, very thick] (do\i) -- (di\i);
|
||||
\draw[blue!70!black, very thick] (di\i) -- (do\j);
|
||||
}
|
||||
% Chord for the bridge (one chord across the dual cycle)
|
||||
\draw[blue!70!black, very thick, dashed] (di0) to[bend left=20] (di3);
|
||||
% Labels
|
||||
\node[red] at (0, \Rout + 0.35) {\small $B_{\mathrm{out}}$};
|
||||
\node[red!60!white] at (0, -\Rin + 0.15) {\small $B_{\mathrm{in}}$};
|
||||
\node[blue!70!black] at (\Rout + 0.85, 0.55) {\small Hamilton walk};
|
||||
\node[blue!70!black] at (\Rout + 0.85, 0.25) {\small + non-crossing};
|
||||
\node[blue!70!black] at (\Rout + 0.85, -0.05) {\small chord};
|
||||
\end{tikzpicture}
|
||||
\caption{Case 2 ($R$ = annulus) with a single ``bridge''-style
|
||||
chord. Outer boundary $B_{\mathrm{out}}$ and inner boundary
|
||||
$B_{\mathrm{in}}$ are concentric hexagons (red). The annular
|
||||
region is triangulated by spokes (grey) and one extra interior
|
||||
annular edge between two inner vertices (dashed grey). The
|
||||
inner dual $\Gamma$ (blue) consists of $12$ dual vertices at the
|
||||
$12$ annular face centroids, connected as a Hamilton cycle around
|
||||
the annulus, plus one chord (dashed blue) corresponding to the
|
||||
extra interior edge. All $12$ vertices lie on the outer face of
|
||||
the chord-augmented cycle, so $\Gamma$ is outerplanar.}
|
||||
\label{fig:inner-dual-annulus-case}
|
||||
\end{figure}
|
||||
|
||||
\begin{remark}
|
||||
\label{rem:hamilton-cycle-spoke-only}
|
||||
In the \emph{spoke-only} case (Definition~\ref{def:tire-graph} with
|
||||
|
||||
Reference in New Issue
Block a user