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coloring_nested_tire_graphs: TikZ figures for T_∂ (thick + thin H_1)

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Two illustrations in boundary_cut_tire.tex:

1. Thick H_1 case: hexagonal H_1 with 3 OUT pendants (outward,
   dashed red) and 3 IN pendants (inward, dashed green). Shows
   T_∂ cycle as the hexagon boundary between low-side outer
   region (red shade) and high-side inner region (green shade).

2. Thin H_1 case: H_1 = path of 3 edges (tree), single face is
   f_∂, boundary walk traverses each edge twice (gray arrows
   showing the two-sided traversal). 2 OUT pendants at V_{deg=2}
   vertices, no IN pendants. This is the dodecahedron cut #0
   side 0 structure where the high-side-only DP failed.

Note grows from 3 to 4 pages.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-26 23:22:05 -04:00
parent 1556109dda
commit 197b64df64
4 changed files with 469 additions and 57 deletions
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf
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@@ -3,6 +3,8 @@
\usepackage{graphicx}
\usepackage{geometry}
\usepackage{booktabs}
\usepackage{tikz}
\usetikzlibrary{backgrounds,decorations.pathreplacing}
\geometry{margin=1in}
\title{The boundary cut tire $T_\partial$: closing the coverage gap}
@@ -74,6 +76,56 @@ $T_\partial^{(i)}$ obtained from:
\end{itemize}
\end{defn}
\paragraph{Picture.} A thick-side example with $H_1$ a hexagonal
cycle in $G'_i$: the low-side face $f_\partial$ is the outer
(unbounded) region containing $3$ pendants (depth $0$); the
adjacent high-side face is inside the hexagon, containing depth-$2$
edges.
\begin{center}
\begin{tikzpicture}[scale=1.6]
\def\R{1.6}
\foreach \i in {0,...,5} {
\pgfmathsetmacro{\ang}{60*\i + 90}
\node[circle, fill=black, inner sep=1.3pt] (v\i) at (\ang:\R) {};
}
\begin{scope}[on background layer]
\fill[red!8] (-3.2,-2.4) rectangle (3.2,2.4);
\fill[green!8]
(90:\R) -- (150:\R) -- (210:\R) -- (270:\R) --
(330:\R) -- (30:\R) -- cycle;
\end{scope}
\foreach \i in {0,...,5} {
\pgfmathtruncatemacro{\j}{mod(\i+1,6)}
\draw[blue, very thick] (v\i) -- (v\j);
}
\foreach \i in {0,2,4} {
\pgfmathsetmacro{\ang}{60*\i + 90}
\coordinate (p\i) at (\ang:\R+0.85);
\draw[red, thick, dashed] (v\i) -- (p\i);
\node[circle, draw=red, fill=white, inner sep=1.2pt] at (p\i) {};
}
\foreach \i in {1,3,5} {
\pgfmathsetmacro{\ang}{60*\i + 90}
\coordinate (q\i) at (\ang:\R-0.7);
\draw[green!55!black, thick, dashed] (v\i) -- (q\i);
\node[circle, fill=green!55!black, inner sep=1.2pt] at (q\i) {};
}
\node[blue] at (0, \R+0.35) {\small cycle of $T_\partial$ ($= \partial f_\partial$, depth $1$)};
\node[red] at (2.95, 1.55) {\small OUT pendants};
\node[red] at (2.95, 1.25) {\small (depth $0$, cut)};
\node[green!55!black] at (0, -0.05) {\small IN pendants (depth $2$)};
\node at (-2.55, -1.95) {\small low-side face $f_\partial$};
\node at (2.4, -2.05) {\small (high-side interior)};
\end{tikzpicture}
\end{center}
The cycle of $T_\partial$ is the hexagon (blue, depth $1$); pendants
fall in two classes: OUT (red, dashed, going into the low-side
face) attach to the cut, and IN (green, dashed, going into the
adjacent high-side region) attach to depth-$2$ edges shared with
$T_2^{(f')}$ tires.
\paragraph{What $T_\partial$ looks like in special cases.}
\begin{itemize}
\item \emph{Side with thick BFS:} $H_1$ has multiple faces, one
@@ -89,6 +141,41 @@ $T_\partial^{(i)}$ obtained from:
No IN pendants (no depth-$2$ edges exist).
\end{itemize}
\begin{center}
\begin{tikzpicture}[scale=1.5]
\node[circle, fill=black, inner sep=1.3pt] (a) at (-1.5, 0) {};
\node[circle, fill=black, inner sep=1.3pt] (b) at (-0.5, 0) {};
\node[circle, fill=black, inner sep=1.3pt] (c) at ( 0.5, 0) {};
\node[circle, fill=black, inner sep=1.3pt] (d) at ( 1.5, 0) {};
\begin{scope}[on background layer]
\fill[red!8] (-3.2,-1.4) rectangle (3.2,1.4);
\end{scope}
\draw[blue, very thick] (a) -- (b) -- (c) -- (d);
\draw[blue, thick, decorate, decoration={brace, amplitude=4pt, raise=8pt}]
(a) -- (d) node[midway, above=12pt, blue] {\small $H_1$ (a tree, 3 edges)};
\coordinate (pb) at (-0.5, -0.9);
\coordinate (pc) at ( 0.5, -0.9);
\draw[red, thick, dashed] (b) -- (pb);
\draw[red, thick, dashed] (c) -- (pc);
\node[circle, draw=red, fill=white, inner sep=1.2pt] at (pb) {};
\node[circle, draw=red, fill=white, inner sep=1.2pt] at (pc) {};
\node[red] at (-2.2, -0.5) {\small OUT pendants};
\node[red] at (-2.2, -0.8) {\small (depth $0$)};
\node at (2.5, 0.4) {\small low-side $f_\partial$};
\node at (2.5, 0.1) {\small (the only face)};
\draw[->, gray, thick]
(-1.5, 0.5) .. controls (-2.2, 1.2) and (2.2, 1.2) .. (1.5, 0.5);
\draw[->, gray, thick]
(1.5, -0.5) .. controls (2.2, -1.2) and (-2.2, -1.2) .. (-1.5, -0.5);
\node[gray] at (0, 1.05) {\small boundary walk traverses each edge twice};
\end{tikzpicture}
\end{center}
In the thin case (e.g.\ dodecahedron cut $\#0$ side $0$ with
$|S_0| = 4$), the boundary walk of the single face has length $6$
(each of $3$ $H_1$ edges visited twice), and OUT pendants attach at
the two $V_{\deg = 2}$ vertices.
\section*{The extended forest}
The high-side cut tire forest of $G'_i$ (proven in