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 06009b0..f712371 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux @@ -1,6 +1,6 @@ \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}{The coverage gap.}{2}{}\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 } 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 884f36a..3a3462d 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 23:33 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 23:36 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: - 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 + 14740 strings out of 478268 + 301161 string characters out of 5846347 + 568755 words of memory out of 5000000 + 32695 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 @@ -617,7 +617,7 @@ ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb> -Output written on boundary_cut_tire.pdf (5 pages, 249957 bytes). +Output written on boundary_cut_tire.pdf (5 pages, 251070 bytes). PDF statistics: 134 PDF objects out of 1000 (max. 8388607) 82 compressed objects within 1 object stream 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..2c487e6 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..28d6ee6 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex @@ -36,39 +36,49 @@ $H_{d-1}$ 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); + % Y (outside H_{d-1}): light yellow + \fill[yellow!22] (-4.4, -3.0) rectangle (4.4, 3.0); + % X (between H_{d-1} and H_d): light cyan + \fill[cyan!18] (0, 0) circle (2.4); + % A (inside H_d): white \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); + \draw[blue, very thick] (0, 0) circle (2.4); + \draw[orange!90!black, very thick] (0, 0) circle (1.0); + % Mark "face B = X ∪ Y" with a dashed red boundary just outside H_d + \draw[red, thick, dashed] (0, 0) circle (1.08); + \node[orange!90!black] at (0, 0) {\small $A$}; + \node[blue] at (1.05, 2.25) {\small $H_{d-1}$}; + \node[orange!90!black] at (-1.05, -0.45) {\small $H_d$}; + \node[fill=cyan!18, inner sep=1pt] at (1.6, 1.35) {\small face $X$ of $H_{d-1}$}; + \node[fill=yellow!22, inner sep=1pt] at (3.2, 2.55) {\small face $Y$ of $H_{d-1}$}; + \node[red] at (-2.95, 1.45) {\small face $B$ of $H_d$}; + \node[red] at (-2.95, 1.15) {\small (= everything}; + \node[red] at (-2.95, 0.85) {\small outside red dashed)}; + \draw[->, red, thick] (-1.95, 1.0) -- (-1.15, 1.0); + \draw[->, red, thick] (-1.95, 1.0) .. controls (-2.3, 1.6) and (-3.1, 2.4) .. (-3.5, 2.7); \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: +Here face $B$ of $H_d$ (low-side) is \emph{everything outside the +dashed red boundary} --- i.e.\ the union of the cyan annulus and the +exterior yellow region. This is a single connected face of +$\mathbb{R}^2 \setminus H_d$ (the red boundary is just the inner +edge of $H_d$; from $B$'s perspective there is no separating curve +out at $H_{d-1}$). + +But the two \emph{faces of $H_{d-1}$} are distinct regions: \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). +\item Face $X$ of $H_{d-1}$: the cyan annulus between $H_d$ and + $H_{d-1}$. It is a proper subset of $B$ (the yellow exterior + lies outside $X$). +\item Face $Y$ of $H_{d-1}$: the yellow exterior region. Also a + proper subset of $B$ (the cyan annulus lies outside $Y$). \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 +So $X \subsetneq B$ and $Y \subsetneq B$, but $B \not\subseteq X$ +and $B \not\subseteq Y$. Neither $X$ nor $Y$ contains all of $B$, +so $B$ has no unique parent face in $H_{d-1}$. This is the ``uniqueness step'' that fails for low-side faces. By contrast, face $A$ (high-side, inside the inner cycle) sits