From 587d92873612e81d871f57430bd4258ddc4a6b35 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 23:27:30 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20add=20nest?= =?UTF-8?q?ed=20cut=20tire=20figure=20to=20T=5F=E2=88=82=20note?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit NEW SECTION: "High-side cut tires alongside T_∂" with a stacked figure showing concentric structure: - Outer hexagon (blue) = T_∂'s cycle (depth 1 = ∂f_∂). - Inner triangle (orange) = a depth-2 high-side cut tire T_2^(f') interior to T_∂. - Red dashed pendants outward from T_∂ = cut edges (depth 0, OUT spokes of T_∂). - Green dashed edges between outer and inner = depth-2 edges, serving simultaneously as IN pendants of T_∂ and OUT spokes of T_2^(f'). These are the SHARED EDGES the chain DP uses. - Purple dashed pendants from inner triangle = depth-3 edges (IN spokes of T_2^(f'), going to grandchildren). The picture makes concrete what was abstract: each tire has the same shape (cycle + 2 classes of spokes), tires nest concentrically by depth, and the chain DP's edge-sharing is literally the geometric "this same edge is on both tires." Note grows from 4 to 5 pages. 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+\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