diff --git a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux index 01e7d6a..06009b0 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux @@ -1,13 +1,14 @@ \relax +\@writefile{toc}{\contentsline {paragraph}{Why low-side faces break uniqueness.}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{The coverage gap.}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Resolution.}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Setup.}{1}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{The low-side face of $H_1$.}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Resolution.}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Setup.}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{The low-side face of $H_1$.}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Picture.}{2}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{What $T_\partial $ looks like in special cases.}{2}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Role in the chain DP.}{3}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{What this closes.}{4}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{What it leaves open.}{4}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Why the chain DP can still work.}{4}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Logical status of the extended framework.}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{What $T_\partial $ looks like in special cases.}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Role in the chain DP.}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{What this closes.}{5}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{What it leaves open.}{5}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Why the chain DP can still work.}{5}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Logical status of the extended framework.}{5}{}\protected@file@percent } \gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log index 9cc0185..884f36a 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 23:27 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 23:33 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -587,10 +587,10 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B [4] [5] (./boundary_cut_tire.aux) ) Here is how much of TeX's memory you used: - 14724 strings out of 478268 - 300879 string characters out of 5846347 - 568561 words of memory out of 5000000 - 32679 multiletter control sequences out of 15000+600000 + 14739 strings out of 478268 + 301145 string characters out of 5846347 + 571661 words of memory out of 5000000 + 32694 multiletter control sequences out of 15000+600000 482413 words of font info for 79 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 84i,6n,89p,452b,713s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -609,16 +609,18 @@ mr7.pfb> - -Output written on boundary_cut_tire.pdf (5 pages, 233743 bytes). + +Output written on boundary_cut_tire.pdf (5 pages, 249957 bytes). PDF statistics: - 124 PDF objects out of 1000 (max. 8388607) - 76 compressed objects within 1 object stream + 134 PDF objects out of 1000 (max. 8388607) + 82 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 13a4735..31c0782 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 fb37633..70cb62f 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex @@ -27,6 +27,54 @@ 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