From 410c2a1b35ffaf6adbe9f06ed904c2ec0f6785a4 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 22:24:29 -0400 Subject: [PATCH] coloring_nested_tire_graphs: detailed proof of the cut tire forest proposition Replaces the earlier sketch with a more detailed two-stage proof: Stage 1: BFS level-set lemma. Lemma (BFS-adj): adjacent edges in line graph differ in depth by at most 1. Proof: BFS-distance property. Lemma (level-set): every face of H_d contains only edges of depth d. Proof: if a face contains both, the line-graph walk connecting them must pass through a depth-d edge (by BFS-adj), contradicting the walk being in the face (= R^2 \ H_d). Stage 2: faces of H_{d+1} embed in faces of H_d. Key claim: no H_d edge sits strictly inside any face of H_{d+1}. Informal topological argument: any H_d edge intruder into f' must already lie on f''s boundary closure. Stage 3: forest structure follows from unique-parent + strictly decreasing depth. HONESTLY ACKNOWLEDGED GAP: the topological argument in Stage 2 is informal; a rigorous proof would set up the planar rotation system and trace boundary walks carefully. Empirically the conclusion holds across 1486 tested cases (0 failures), giving very strong support. Note grows to 5 pages. Co-Authored-By: Claude Opus 4.7 --- .../notes/cut_tire_tree_structure.aux | 7 +- .../notes/cut_tire_tree_structure.log | 86 ++++++++--- .../notes/cut_tire_tree_structure.pdf | Bin 196689 -> 210419 bytes .../notes/cut_tire_tree_structure.tex | 135 ++++++++++++++++-- 4 files changed, 197 insertions(+), 31 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux index 6d971df..d2df511 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux @@ -1,4 +1,7 @@ \relax \newlabel{prop:tree}{{}{1}} -\@writefile{toc}{\contentsline {paragraph}{Reformulated chain half (tree DP form).}{4}{}\protected@file@percent } -\gdef \@abspage@last{4} +\newlabel{lem:bfs-adj}{{}{1}} +\newlabel{lem:level-set}{{}{1}} +\@writefile{toc}{\contentsline {paragraph}{Caveat on Stage 2.}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Reformulated chain half (tree DP form).}{5}{}\protected@file@percent } +\gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log index 823b541..33e26fc 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.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 22:15 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 22:24 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -260,28 +260,81 @@ LaTeX Font Info: Trying to load font information for U+msb on input line 16. 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The relation defines a forest. +\begin{lem}[BFS depth differs by at most 1 between adjacent edges] +\label{lem:bfs-adj} +Let $e_1, e_2 \in E(G'_i)$ share a vertex (so they are adjacent in +the line graph). Then $|\mathrm{depth}(e_1) - \mathrm{depth}(e_2)| +\le 1$. +\end{lem} -The roots are the depth-$1$ cut tires. Their ``virtual parent'' is -the depth-$0$ pendant configuration, i.e.\ the cut $C$ itself. +\begin{proof} +By definition of BFS depth, $\mathrm{depth}(e) = $ minimum line-graph +distance from $e$ to any pendant. If $e_1, e_2$ are line-graph +adjacent, then a shortest line-graph path from a pendant to $e_2$ +can be extended by the one step from $e_2$ to $e_1$, yielding a path +of length $\mathrm{depth}(e_2) + 1$ from a pendant to $e_1$. So +$\mathrm{depth}(e_1) \le \mathrm{depth}(e_2) + 1$, and symmetrically. \end{proof} +\begin{lem}[Level-set property of $H_d$] +\label{lem:level-set} +For each $d \ge 1$, every face of $H_d$ in the inherited planar +embedding satisfies one of the following: +\begin{itemize} + \item Every edge of $G'_i$ strictly inside the face has depth $< d$ + (a ``low-side'' face), or + \item Every edge of $G'_i$ strictly inside the face has depth $> d$ + (a ``high-side'' face). +\end{itemize} +\end{lem} + +\begin{proof} +Let $f$ be a face of $H_d$. Suppose for contradiction that $f$ +contains an edge $e_a$ of depth $a < d$ and an edge $e_b$ of depth +$b > d$ strictly inside. Since $f$ is a connected open region, +there is a continuous path in $f$ from a point on $e_a$ to a point +on $e_b$ avoiding $H_d$'s edges (since $f \subseteq +\mathbb{R}^2 \setminus H_d$). + +Slightly perturbed, this path is realised as a sequence of edges in +$G'_i \setminus H_d$ together with possibly some vertices in +$V(G'_i)$ shared between consecutive edges --- i.e.\ a line-graph +walk in $G'_i \setminus H_d$ from $e_a$ to $e_b$ that stays inside +$\overline{f}$. + +By Lemma~\ref{lem:bfs-adj}, consecutive edges along this line-graph +walk differ in depth by at most $1$. Going from depth $a < d$ to +depth $b > d$, the walk must pass through some edge of depth exactly +$d$. But that edge is in $H_d$, contradicting that the walk lies in +$G'_i \setminus H_d$. + +Hence $f$ contains only edges of depth $< d$, or only edges of depth +$> d$ (or neither, if $f$ contains no edges of $G'_i$ in its +interior). +\end{proof} + +\medskip +\noindent\textbf{Stage 2: faces of $H_{d+1}$ embed in faces of $H_d$.} + +Pendants (depth $0$ edges) lie in some specific face of $H_d$; that +face is low-side. All other faces of $H_d$ are high-side and +contain depth-$> d$ edges, which includes all of $H_{d+1}$'s edges. + +Let $f'$ be a face of $H_{d+1}$. We claim $f'$ is contained in +exactly one face of $H_d$. + +\emph{Containment in at least one face:} $f'$ is an open connected +region of $\mathbb{R}^2 \setminus H_{d+1}$. In particular it is +connected. By Lemma~\ref{lem:level-set}, each face of $H_d$ is +either entirely low-side or entirely high-side, and the two types +are separated topologically by $H_d$. Suppose for contradiction +$f'$ intersects two distinct faces $g_1, g_2$ of $H_d$. Then a +path in $f'$ from a point in $g_1$ to a point in $g_2$ crosses some +edge of $H_d$ (since faces of $H_d$ are separated by $H_d$ edges). +But $H_d \subset E(G'_i) \setminus E(H_{d+1})$, so $H_d$ edges are +in $\mathbb{R}^2 \setminus E(H_{d+1})$; they could in principle lie +within $f'$ \emph{except} that $f'$ is a maximal connected open +component of that complement, which already includes the $H_d$ +edges. This is where the elementary topological argument is +subtle: we need the additional constraint that no $H_d$ edge +sits strictly inside $f'$. + +\emph{No $H_d$ edge sits strictly inside $f'$:} suppose an $H_d$ edge +$e$ is strictly inside $f'$. Then $e$'s endpoints are inside $f'$ +(or on $\partial f'$). An endpoint $v$ of $e$ is also incident to +$H_{d+1}$ edges (since $V(H_d) \cap V(H_{d+1})$ contains vertices +where depth-$d$ and depth-$(d+1)$ edges meet; in cubic $G'_i$, $v$ +has $3$ edges with various depths). The $H_{d+1}$ edges incident +to $v$ are on $\partial f'$ (the boundary walk of $f'$), so $v \in +\partial f'$. Then $e$'s other endpoint $w$ is also on or inside +$f'$. But moving from $v$ along $e$ into $w$: this curve segment +is inside $f'$ until it reaches $w$. If $w$ is on $\partial f'$, +the entire edge $e$ lies on the boundary closure $\overline{f'}$, +not strictly inside. If $w$ is strictly inside $f'$, then $w$'s +incident edges (including $e$) project into $f'$ in a way that +should appear on $\partial f'$ --- but $e$ is not in $H_{d+1}$, +contradiction. + +\medskip +The careful case analysis shows: no $H_d$ edge sits strictly inside +$f'$, hence $f'$ is contained in a single face of $H_d$ (the unique +face whose interior contains $f'$). + +\medskip +\noindent\textbf{Conclusion: forest structure.} + +The parent relation $(d+1, f') \mapsto (d, f)$ assigns each +$H_{d+1}$ face $f'$ to a unique $H_d$ face $f$ containing it. Since +parent depth is strictly less than child depth, walking up parent +links strictly decreases depth, terminating at a depth-$1$ root (or +at the ``cut'' for the depth-$1$ roots' virtual parent). No cycles +can form. Hence the parent relation defines a forest. \qed +\end{proof} + +\paragraph{Caveat on Stage 2.} The argument that ``no $H_d$ edge sits +strictly inside $f'$'' uses an informal topological case analysis on +how an $H_d$ edge inside $f'$ would have to interact with $f'$'s +boundary. A fully rigorous proof would set up the topological +framework more carefully (e.g.\ via the rotation system of the +planar embedding, tracing the boundary walk of $f'$ around an +``intruder'' $H_d$ edge to show it must already lie in +$\partial f'$). Empirically, the conclusion holds across +\textbf{$1486$ tested cases, $0$ failures} (see broader sweep below). + \section*{Why this matters for the chain half} Chain pigeonhole asks whether the per-tire $S_3$-orbit structure