didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Revert "coloring_nested_tire_graphs: clarify uniqueness-failure figure"

Browse Source 
This reverts commit 6c6d1eac94.
This commit is contained in:
didericis
2026-05-26 23:48:54 -04:00
parent 32dd4d3553
commit b2e1239c06
4 changed files with 33 additions and 43 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux
+1 -1
View File
@@ -1,6 +1,6 @@
\relax \relax
\@writefile{toc}{\contentsline {paragraph}{Why low-side faces break uniqueness.}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Why low-side faces break uniqueness.}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{The coverage gap.}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{The coverage gap.}{1}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Resolution.}{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}{Setup.}{2}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{The low-side face of $H_1$.}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{The low-side face of $H_1$.}{2}{}\protected@file@percent }
papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log
+6 -6
View File
@@ -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 23:36 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 entering extended mode
restricted \write18 enabled. restricted \write18 enabled.
%&-line parsing enabled. %&-line parsing enabled.
@@ -587,10 +587,10 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B
[4] [5] [4] [5]
(./boundary_cut_tire.aux) ) (./boundary_cut_tire.aux) )
Here is how much of TeX's memory you used: Here is how much of TeX's memory you used:
14740 strings out of 478268 14739 strings out of 478268
301161 string characters out of 5846347 301145 string characters out of 5846347
568755 words of memory out of 5000000 571661 words of memory out of 5000000
32695 multiletter control sequences out of 15000+600000 32694 multiletter control sequences out of 15000+600000
482413 words of font info for 79 fonts, out of 8000000 for 9000 482413 words of font info for 79 fonts, out of 8000000 for 9000
1141 hyphenation exceptions out of 8191 1141 hyphenation exceptions out of 8191
84i,6n,89p,452b,713s stack positions out of 10000i,1000n,20000p,200000b,200000s 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></usr/loc
al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/loca al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/loca
l/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb></usr/ l/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb></usr/
local/texlive/2022/texmf-dist/fonts/type1/public/cm-super/sfrm1095.pfb> local/texlive/2022/texmf-dist/fonts/type1/public/cm-super/sfrm1095.pfb>
Output written on boundary_cut_tire.pdf (5 pages, 251070 bytes). Output written on boundary_cut_tire.pdf (5 pages, 249957 bytes).
PDF statistics: PDF statistics:
134 PDF objects out of 1000 (max. 8388607) 134 PDF objects out of 1000 (max. 8388607)
82 compressed objects within 1 object stream 82 compressed objects within 1 object stream
papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf
BIN
View File
Binary file not shown.
papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex
+26 -36
View File
@@ -36,49 +36,39 @@ $H_{d-1}$ edges).
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.95] \begin{tikzpicture}[scale=0.95]
\begin{scope}[on background layer]
% 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}
\draw[blue, very thick] (0, 0) circle (2.4); \draw[blue, very thick] (0, 0) circle (2.4);
\draw[orange!90!black, very thick] (0, 0) circle (1.0); \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 \begin{scope}[on background layer]
\draw[red, thick, dashed] (0, 0) circle (1.08); \fill[red!10] (-4.2, -3.0) rectangle (4.2, 3.0);
\node[orange!90!black] at (0, 0) {\small $A$}; \fill[white] (0, 0) circle (1.0);
\node[blue] at (1.05, 2.25) {\small $H_{d-1}$}; \end{scope}
\node[orange!90!black] at (-1.05, -0.45) {\small $H_d$}; \node[blue] at (-1.05, 2.25) {\small $H_{d-1}$};
\node[fill=cyan!18, inner sep=1pt] at (1.6, 1.35) {\small face $X$ of $H_{d-1}$}; \node[orange!90!black] at (-0.75, -0.55) {\small $H_d$};
\node[fill=yellow!22, inner sep=1pt] at (3.2, 2.55) {\small face $Y$ of $H_{d-1}$}; \node[blue, fill=white, inner sep=1.5pt] at (0, 0.4) {\small face $A$};
\node[red] at (-2.95, 1.45) {\small face $B$ of $H_d$}; \node[blue, fill=white, inner sep=1.5pt] at (0, -0.0) {\small (high-side)};
\node[red] at (-2.95, 1.15) {\small (= everything}; \node[red] at (-3.4, 0.2) {\small face $B$ of $H_d$};
\node[red] at (-2.95, 0.85) {\small outside red dashed)}; \node[red] at (-3.4, -0.1) {\small (low-side)};
\draw[->, red, thick] (-1.95, 1.0) -- (-1.15, 1.0); \node[gray] at (2.3, 0.0) {\small face $X$ of $H_{d-1}$};
\draw[->, red, thick] (-1.95, 1.0) .. controls (-2.3, 1.6) and (-3.1, 2.4) .. (-3.5, 2.7); \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{tikzpicture}
\end{center} \end{center}
Here face $B$ of $H_d$ (low-side) is \emph{everything outside the The red region is face $B$ (low-side of $H_d$): a single connected
dashed red boundary} --- i.e.\ the union of the cyan annulus and the region of $\mathbb{R}^2 \setminus H_d$. But $B$ is split by the
exterior yellow region. This is a single connected face of $H_{d-1}$ cycle into:
$\mathbb{R}^2 \setminus H_d$ (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} \begin{itemize}
\item Face $X$ of $H_{d-1}$: the cyan annulus between $H_d$ and \item Face $X$ of $H_{d-1}$: the annular region between $H_d$
$H_{d-1}$. It is a proper subset of $B$ (the yellow exterior and $H_{d-1}$.
lies outside $X$). \item Face $Y$ of $H_{d-1}$: the exterior of $H_{d-1}$ (where the
\item Face $Y$ of $H_{d-1}$: the yellow exterior region. Also a pendants live).
proper subset of $B$ (the cyan annulus lies outside $Y$).
\end{itemize} \end{itemize}
So $X \subsetneq B$ and $Y \subsetneq B$, but $B \not\subseteq X$ Neither $X$ nor $Y$ contains \emph{all} of $B$ --- they each contain
and $B \not\subseteq Y$. Neither $X$ nor $Y$ contains all of $B$, a piece. If we tried to make $B$ a child of some unique
so $B$ has no unique parent face in $H_{d-1}$. This is the $H_{d-1}$ face in the tree, no such parent exists. This is the
``uniqueness step'' that fails for low-side faces. ``uniqueness step'' that fails for low-side faces.
By contrast, face $A$ (high-side, inside the inner cycle) sits By contrast, face $A$ (high-side, inside the inner cycle) sits