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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LaTeX Error: Environment lem undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.57 \begin{lem} + [Level-set property of $H_d$] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: \begin{proof} on input line 35 ended by \end{lem}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.67 \end{lem} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +[1 + +{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] +Overfull \hbox (83.47609pt too wide) in paragraph at lines 175--179 []\OT1/cmr/m/n/10.95 Compatibility with each child $\OML/cmm/m/it/10.95 T[]$ \O T1/cmr/m/n/10.95 via the bi-jec-tion $\OMS/cmsy/m/n/10.95 f[]\OML/cmm/m/it/10.9 5 T[]\OMS/cmsy/m/n/10.95 g $ f[]\OML/cmm/m/it/10.95 T[]\OMS/cmsy/m/n/10.95 g$\O T1/cmr/m/n/10.95 . 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PDF statistics: - 98 PDF objects out of 1000 (max. 8388607) - 59 compressed objects within 1 object stream + 106 PDF objects out of 1000 (max. 8388607) + 64 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 1 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.pdf b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.pdf index a3161e7..512fa95 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.pdf and b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.tex b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.tex index 56c149b..48d5332 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.tex +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.tex @@ -32,24 +32,133 @@ The forest's roots are the cut tires at depth $1$ (one per face of $H_1$); their ``virtual parent'' is the cut $C$ itself. \end{prop} -\begin{proof}[Sketch] -$H_{d+1}$ is a subgraph of $G'_i$ with the inherited planar -embedding. Each face of $H_{d+1}$ is a maximal connected open -region of $|\Pi| \setminus E(H_{d+1})$ in the plane. +\begin{proof} +We prove the proposition in two stages. -In particular, every face of $H_{d+1}$ lies inside some face of $H_d$ -(since $H_d$ has fewer edges and so larger faces). ``Lies inside'' -means: the open face region of $H_{d+1}$ is a subset of an open face -region of $H_d$. This containment is unique because the faces of -$H_d$ partition $|\Pi| \setminus E(H_d)$. +\medskip +\noindent\textbf{Stage 1: the BFS level-set lemma.} -Hence parent is well-defined and unique. No face of $H_{d+1}$ is -its own parent (because $d+1 > d$). 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