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 db775d9..01e7d6a 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux @@ -6,8 +6,8 @@ \@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.}{3}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{What it leaves open.}{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 } -\gdef \@abspage@last{4} +\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 65bf56b..9cc0185 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 23:21 +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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -584,16 +584,16 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) [1 {/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] - [4] + [4] [5] (./boundary_cut_tire.aux) ) Here is how much of TeX's memory you used: - 14594 strings out of 478268 - 299023 string characters out of 5846347 - 567504 words of memory out of 5000000 - 32549 multiletter control sequences out of 15000+600000 + 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 482413 words of font info for 79 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 - 84i,6n,89p,452b,712s stack positions out of 10000i,1000n,20000p,200000b,200000s + 84i,6n,89p,452b,713s stack positions out of 10000i,1000n,20000p,200000b,200000s {/usr/local/texlive/2022/texmf-dist/fonts/enc/dvips/ cm-super/cm-super-ts1.enc} -< -/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti7.pfb> -Output written on boundary_cut_tire.pdf (4 pages, 220265 bytes). +pfb> + +Output written on boundary_cut_tire.pdf (5 pages, 233743 bytes). PDF statistics: - 116 PDF objects out of 1000 (max. 8388607) - 71 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 3e77a03..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 296800b..fb37633 100644 --- a/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex +++ b/papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex @@ -176,6 +176,82 @@ $|S_0| = 4$), the boundary walk of the single face has length $6$ (each of $3$ $H_1$ edges visited twice), and OUT pendants attach at the two $V_{\deg = 2}$ vertices. +\section*{High-side cut tires alongside $T_\partial$} + +A high-side cut tire $T_d^{(f)}$ at depth $d$ has the same +structural shape as $T_\partial$: +\begin{itemize} +\item Cycle = depth-$d$ edges in $\partial f$ ($H_d$ edges). +\item OUT spokes = depth-$(d-1)$ edges at boundary vertices + (pointing outward, toward parent). +\item IN spokes = depth-$(d+1)$ edges at boundary vertices + (pointing inward, toward children). +\end{itemize} + +\begin{center} +\begin{tikzpicture}[scale=1.3] + \def\Rout{2.1} + \def\Rin{1.0} + % Outer hexagon = T_∂ cycle (depth 1) + \foreach \i in {0,...,5} { + \pgfmathsetmacro{\ang}{60*\i + 90} + \node[circle, fill=black, inner sep=1.2pt] (vo\i) at (\ang:\Rout) {}; + } + % Inner triangle = T_2^{(f')} cycle (depth 2) + \foreach \i in {0,1,2} { + \pgfmathsetmacro{\ang}{120*\i + 90} + \node[circle, fill=black, inner sep=1.2pt] (vi\i) at (\ang:\Rin) {}; + } + % Outer cycle edges (T_∂) + \foreach \i in {0,...,5} { + \pgfmathtruncatemacro{\j}{mod(\i+1,6)} + \draw[blue, very thick] (vo\i) -- (vo\j); + } + % Inner cycle edges (T_2) + \foreach \i in {0,1,2} { + \pgfmathtruncatemacro{\j}{mod(\i+1,3)} + \draw[orange!85!black, very thick] (vi\i) -- (vi\j); + } + % OUT pendants from outer hexagon (depth 0, cut) at v1, v3, v5 + \foreach \i in {1,3,5} { + \pgfmathsetmacro{\ang}{60*\i + 90} + \coordinate (p\i) at (\ang:\Rout+0.65); + \draw[red, thick, dashed] (vo\i) -- (p\i); + \node[circle, draw=red, fill=white, inner sep=1pt] at (p\i) {}; + } + % Shared spokes between outer hexagon and inner triangle: + % depth-2 edges = T_∂'s IN pendants AND T_2's OUT spokes + \draw[green!55!black, thick, dashed] (vo0) -- (vi0); + \draw[green!55!black, thick, dashed] (vo2) -- (vi1); + \draw[green!55!black, thick, dashed] (vo4) -- (vi2); + % T_2's IN pendants at inner vertices (depth 3) + \foreach \i in {0,1,2} { + \pgfmathsetmacro{\ang}{120*\i + 90 + 60} + \coordinate (qi\i) at ([shift={(\ang:0.45)}]vi\i); + \draw[purple, thick, dashed] (vi\i) -- (qi\i); + \node[circle, fill=purple, inner sep=1pt] at (qi\i) {}; + } + % Annotations + \node[blue] at (-2.65, 1.75) {\small $T_\partial$ cycle}; + \node[blue] at (-2.65, 1.45) {\small (depth 1)}; + \node[orange!85!black] at (-1.85, -0.05) {\small $T_2^{(f')}$}; + \node[orange!85!black] at (-1.85, -0.35) {\small (depth 2)}; + \node[red] at (2.75, 1.65) {\small cut}; + \node[red] at (2.75, 1.35) {\small (depth 0)}; + \node[green!55!black] at (2.75, -1.35) {\small shared:}; + \node[green!55!black] at (2.75, -1.65) {\small IN of $T_\partial$}; + \node[green!55!black] at (2.75, -1.95) {\small $=$ OUT of $T_2$}; + \node[purple] at (0, -0.85) {\small depth 3}; +\end{tikzpicture} +\end{center} + +The green dashed edges are the \emph{shared} depth-$2$ edges: +they are IN pendants of $T_\partial$ \emph{and} OUT spokes of +$T_2^{(f')}$. In any global $3$-edge coloring of $G'_i$, the +color assigned to each green edge is the same when viewed from +either tire. The chain DP exploits exactly this shared-edge +identity. + \section*{The extended forest} The high-side cut tire forest of $G'_i$ (proven in