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PDF statistics: - 134 PDF objects out of 1000 (max. 8388607) - 82 compressed objects within 1 object stream + 124 PDF objects out of 1000 (max. 8388607) + 76 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 13 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf index 31c0782..13a4735 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf and b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex index 70cb62f..fb37633 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex @@ -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 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 (\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