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coloring_nested_tire_graphs: add nested cut tire figure to T_∂ note

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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.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-26 23:27:30 -04:00
parent 197b64df64
commit 587d928736
4 changed files with 97 additions and 20 deletions
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.aux
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\@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 }
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.log
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.pdf
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papers/coloring_nested_tire_graphs/notes/boundary_cut_tire.tex
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@@ -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