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Revert "coloring_nested_tire_graphs: figure showing low-side face uniqueness failure"

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This reverts commit a8af5612c4.
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didericis
2026-05-26 23:48:54 -04:00
parent b2e1239c06
commit 0f57b62d77
4 changed files with 23 additions and 74 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.log
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex
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@@ -27,54 +27,6 @@ pendants of $G'_i$. This omission is essential to prove the
forest's tree structure (low-side faces span multiple parent faces forest's tree structure (low-side faces span multiple parent faces
of $H_{d-1}$, violating the uniqueness step in the proof). of $H_{d-1}$, violating the uniqueness step in the proof).
\paragraph{Why low-side faces break uniqueness.} Concretely:
suppose $H_{d-1}$ is the outer (larger) cycle and $H_d$ a nested
cycle inside. The two faces of $H_d$ are face $A$ (inside the
inner cycle, high-side, contains depth-$>d$ stuff) and face $B$
(outside the inner cycle, low-side, contains pendants and
$H_{d-1}$ edges).
\begin{center}
\begin{tikzpicture}[scale=0.95]
\draw[blue, very thick] (0, 0) circle (2.4);
\draw[orange!90!black, very thick] (0, 0) circle (1.0);
\begin{scope}[on background layer]
\fill[red!10] (-4.2, -3.0) rectangle (4.2, 3.0);
\fill[white] (0, 0) circle (1.0);
\end{scope}
\node[blue] at (-1.05, 2.25) {\small $H_{d-1}$};
\node[orange!90!black] at (-0.75, -0.55) {\small $H_d$};
\node[blue, fill=white, inner sep=1.5pt] at (0, 0.4) {\small face $A$};
\node[blue, fill=white, inner sep=1.5pt] at (0, -0.0) {\small (high-side)};
\node[red] at (-3.4, 0.2) {\small face $B$ of $H_d$};
\node[red] at (-3.4, -0.1) {\small (low-side)};
\node[gray] at (2.3, 0.0) {\small face $X$ of $H_{d-1}$};
\node[gray] at (2.3, -0.3) {\small (between cycles)};
\node[gray] at (3.1, 2.45) {\small face $Y$ of $H_{d-1}$};
\node[gray] at (3.1, 2.15) {\small (outside)};
\draw[->, gray, thick] (2.45, 2.25) -- (2.0, 1.85);
\draw[->, gray, thick] (1.8, 0.0) -- (1.55, -0.0);
\end{tikzpicture}
\end{center}
The red region is face $B$ (low-side of $H_d$): a single connected
region of $\mathbb{R}^2 \setminus H_d$. But $B$ is split by the
$H_{d-1}$ cycle into:
\begin{itemize}
\item Face $X$ of $H_{d-1}$: the annular region between $H_d$
and $H_{d-1}$.
\item Face $Y$ of $H_{d-1}$: the exterior of $H_{d-1}$ (where the
pendants live).
\end{itemize}
Neither $X$ nor $Y$ contains \emph{all} of $B$ --- they each contain
a piece. If we tried to make $B$ a child of some unique
$H_{d-1}$ face in the tree, no such parent exists. This is the
``uniqueness step'' that fails for low-side faces.
By contrast, face $A$ (high-side, inside the inner cycle) sits
entirely inside face $X$ of $H_{d-1}$. Unique parent. This is
why the forest proposition restricts to high-side faces.
\paragraph{The coverage gap.} Empirically \paragraph{The coverage gap.} Empirically
(\texttt{chain\_dp\_joint.py} on the dodecahedron, cut $\#0$, side (\texttt{chain\_dp\_joint.py} on the dodecahedron, cut $\#0$, side
$0$): when $|S_i|$ is small, $H_1$ on side $i$ can be a $0$): when $|S_i|$ is small, $H_1$ on side $i$ can be a