From b526d204ff98c9685bbb109c6a094916bfb06fe4 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 00:16:22 -0400 Subject: [PATCH] coloring_nested_tire_graphs: cyclic-edge-conn distinction + even-cycle question MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit UPDATED: birkhoff_internally_6_connected.tex now adds the distinction between "internally 6-connected" (= cyclic edge conn ≥ 6 in dual) and the framework's needed condition (= cyclic edge conn EXACTLY 6, so 6-edge cuts exist). Notes that this is a real a priori restriction not provided by Birkhoff alone. NEW NOTE: even_separating_cycle.tex (3 pages) Addresses: "must a min 4CT counterexample have a separating n-cycle with n even and n ≥ 6?" Honest answer: I don't know of a proof either way. Key contributions: - Lemma (cut-parity in cubic graphs): |C| ≡ |S| ≡ |T| (mod 2). So even-length cycles in primal G ↔ cuts with even-sized sides in dual G^*. - |V(G^*)| = 2|V(G)| - 4 is always even, so both sides have matching parity. - Birkhoff doesn't rule out odd-length separating cycles ≥ 7. - Second-link heuristic: in internally 6-conn triangulations, the "second link" of any vertex is typically a 6-cycle, giving abundant separating 6-cycles in practice. But this is heuristic, not proven for all such triangulations. Conjecture (stated, not proven): every internally 6-conn planar triangulation with ≥ 12 vertices has a separating even n-cycle with n ≥ 6. Equivalent: every planar cubic graph with cyclic edge connectivity ≥ 6 and ≥ 20 vertices has a cyclic edge cut of size exactly 6. This is a structural question; I don't know a planar cubic counterexample. Co-Authored-By: Claude Opus 4.7 --- .../notes/birkhoff_internally_6_connected.aux | 2 + .../notes/birkhoff_internally_6_connected.log | 61 ++-- .../notes/birkhoff_internally_6_connected.pdf | Bin 161512 -> 182145 bytes .../notes/birkhoff_internally_6_connected.tex | 59 +++- .../notes/even_separating_cycle.aux | 10 + .../notes/even_separating_cycle.log | 297 ++++++++++++++++++ .../notes/even_separating_cycle.pdf | Bin 0 -> 169383 bytes .../notes/even_separating_cycle.tex | 188 +++++++++++ 8 files changed, 583 insertions(+), 34 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/even_separating_cycle.log create mode 100644 papers/coloring_nested_tire_graphs/notes/even_separating_cycle.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/even_separating_cycle.tex diff --git a/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.aux b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.aux index a79beaf..1669de9 100644 --- a/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.aux +++ b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.aux @@ -1,2 +1,4 @@ \relax +\@writefile{toc}{\contentsline {paragraph}{Hidden assumption in the cut-tire framework.}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Empirically.}{3}{}\protected@file@percent } \gdef \@abspage@last{3} diff --git a/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.log b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.log index eef8819..8e657dd 100644 --- a/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.log +++ b/papers/coloring_nested_tire_graphs/notes/birkhoff_internally_6_connected.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) 27 MAY 2026 00:00 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:16 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -562,40 +562,49 @@ LaTeX Font Info: Trying to load font information for U+msb on input line 21. 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+\documentclass[11pt]{article} +\usepackage{amsmath,amssymb,amsthm} +\usepackage{graphicx} +\usepackage{geometry} +\usepackage{booktabs} +\geometry{margin=1in} + +\title{Must a minimum $4$CT counterexample have a separating +$n$-cycle with $n$ even and $n \ge 6$?} +\author{} +\date{} + +\newtheorem*{thm}{Theorem} +\newtheorem*{lem}{Lemma} +\newtheorem*{prop}{Proposition} +\newtheorem*{conj}{Conjecture} +\newtheorem*{rem}{Remark} + +\begin{document} +\maketitle + +\section*{The question} + +\paragraph{Statement.} Let $G$ be a hypothetical minimum +$4$-colour counterexample (a minimum planar triangulation +requiring $\ge 5$ colours). Must $G$ contain a separating +$n$-cycle with $n$ \emph{even} and $n \ge 6$? + +\paragraph{Short answer.} I do not know of a proof either way. +The question is subtle: a parity computation (below) shows that +cuts in the cubic dual $G^*$ have a definite parity tied to the +side sizes, but I do not see this parity forcing the existence of +even-length separating cycles. + +\section*{Parity computation in the cubic dual} + +Let $G$ be a planar triangulation and $G^*$ its cubic planar dual. +$G$ has $V$ vertices, $3V - 6$ edges, $2V - 4$ faces; $G^*$ has +$2V - 4$ vertices, $3V - 6$ edges, $V$ faces. In particular, +$|V(G^*)| = 2V - 4$ is \emph{always even}. + +\begin{lem}[Cut size parity in cubic graphs] +\label{lem:cut-parity} +Let $G^*$ be a cubic graph and let $C$ be an edge cut separating +$V(G^*)$ into sides $S$ and $T = V(G^*) \setminus S$. Then +\[ +|C| \equiv |S| \equiv |T| \pmod{2}. +\] +\end{lem} + +\begin{proof} +Counting degree on side $S$: $3|S| = 2 e_S + |C|$, where $e_S$ is +the number of edges with both endpoints in $S$. Hence +$|C| = 3|S| - 2e_S \equiv |S| \pmod{2}$. Since +$|S| + |T| = |V(G^*)|$ is even, $|S| \equiv |T| \pmod 2$. +\end{proof} + +\paragraph{Translating to primal cycles.} Via the $G \leftrightarrow +G^*$ duality, an $n$-cycle in $G$ corresponds to an $n$-edge cut in +$G^*$. Lemma~\ref{lem:cut-parity} says: an \emph{even} $n$-cycle +in $G$ ($n \ge 6$) corresponds to a $G^*$ cut with both sides +having an \emph{even} number of vertices. An \emph{odd} +$n$-cycle in $G$ ($n \ge 7$) corresponds to both sides having an +\emph{odd} number of vertices. + +\section*{What Birkhoff gives us} + +For a minimum $4$CT counterexample $G$: +\begin{itemize} +\item $G$ is internally $6$-connected (no separating $3$-cycle, no + separating $4$-cycle, no separating $5$-cycle with $\ge 2$ + vertices on each side). +\item Equivalently in $G^*$: cyclic edge connectivity $\ge 6$. +\end{itemize} + +So all sufficiently small cuts in $G^*$ are excluded. The smallest +non-trivial cuts can be $6$-edge ($n = 6$, even) or $7$-edge +($n = 7$, odd), and Birkhoff alone permits both. + +\section*{A potential ``no'' --- can $G^*$ have only odd cuts $\ge 7$?} + +In principle, the minimum non-trivial cut could be $7$ (and all +non-trivial cuts could be odd-length). In this case the +minimum primal separating cycle has length $7$ (odd), and the +question's answer is ``no.'' + +\paragraph{Is this realisable?} We need a planar cubic graph $G^*$ +satisfying: +\begin{itemize} +\item cyclic edge connectivity $\ge 7$; +\item all non-trivial cyclic edge cuts have odd size. +\end{itemize} + +Lemma~\ref{lem:cut-parity} gives a constraint: if a $G^*$ has +\emph{any} cut at all of size $\ge 6$, its parity is fixed by side +size. For ``all cuts odd'' to hold, no even-sized cut would +separate. In a cubic graph this corresponds to the side sizes +being \emph{odd}. Since $|V(G^*)|$ is even (for any planar +triangulation dual), the sides $(|S|, |V(G^*)| - |S|)$ have +matching parity --- both even or both odd. ``All cuts odd'' +means ``all cyclic separations have odd side sizes,'' which is +possible in principle. + +\paragraph{However:} I don't know of a planar cubic graph that has +\emph{only} odd cyclic cuts. The known internally $6$-connected +examples (icosahedron's dual = dodecahedron, etc.) all have +many even cuts of size $6$ as well as odd cuts. + +\section*{A heuristic suggesting ``yes'' --- vertex links} + +Every vertex $v$ in a planar triangulation $G$ has a \emph{link}: +the cycle formed by its neighbours. By Birkhoff, this link is a +$5$-cycle isolating $v$. In $G^*$, this corresponds to a +$5$-edge cut isolating a single triangle face of $G^*$ (= one +vertex of $G$). + +\paragraph{Next layer.} Consider the ``second link'' of $v$: the +set of vertices at $G$-distance exactly $2$ from $v$. This forms +a cycle around the link, of length depending on the degrees of +link vertices. + +If all link vertices have degree $5$ (= minimum for internally +$6$-connected): the link's $5$ vertices contribute $5 \cdot 5 = 25$ +incidences, of which $5$ are to $v$ (the centre) and $2 \cdot 5 += 10$ are between link vertices (the $5$-cycle). The remaining +$25 - 5 - 10 = 10$ incidences go to second-link vertices. But the +$5$ link vertices share their second-link vertices in pairs (each +triangle face containing $v$, a link vertex, and a second-link +vertex), so the second link has $\le 5$ distinct vertices. + +Working through Euler more carefully: if all degrees are $\ge 5$ +and the link of $v$ has length $5$, the second link has length +exactly $5 \cdot (5 - 4) = 5$ (in the icosahedron, second link = +link of antipodal vertex). But in larger triangulations +(degrees of link vertices $\ge 5$, some higher), the second link +is generically a cycle of length $\sum_{u \in \text{link}}(\deg(u) +- 4) = 5 + \sum_u(\deg u - 5) \ge 5$ with equality only when all +link degrees are exactly $5$ (icosahedron case). + +So in larger internally $6$-connected triangulations, second-link +length $\ge 6$, often equal to $6$, often even (especially in +``vertex-transitive enough'' graphs). This is a heuristic for why +$6$-cycle separators are abundant, but it's not a proof. + +\section*{What I can conclude} + +\begin{itemize} +\item \textbf{Parity is determined by side size} + (Lemma~\ref{lem:cut-parity}). Even $n$-cycle separators in $G$ + correspond to even-sided cuts in $G^*$. +\item \textbf{Birkhoff doesn't rule out odd cuts.} Minimum + non-trivial cyclic cut in $G^*$ could in principle be of any + size $\ge 6$. +\item \textbf{No known proof that even $n \ge 6$ separators must + exist} in min $4$CT counterexamples. +\item \textbf{Empirically}, in all tested internally $6$-connected + planar triangulations (icosahedron, pentakis dodecahedron, + Holton--McKay duals), even $6$-cycle separators with + both sides $\ge 2$ vertices exist in abundance. +\end{itemize} + +\section*{Conjecture} + +\begin{conj} +Every internally $6$-connected planar triangulation $G$ with +$|V(G)| \ge 12$ has a separating $n$-cycle with $n$ even and +$n \ge 6$. +\end{conj} + +Equivalently: every planar cubic graph with cyclic edge +connectivity $\ge 6$ and $|V| \ge 20$ has a cyclic edge cut of +size $6$. + +This conjecture seems plausible based on the second-link +heuristic, but I don't have a proof. A planar cubic graph that +violates it would be a structural curiosity worth a name --- a +``cyclically $7$-edge-connected planar cubic graph'' --- and I do +not know an example. + +\paragraph{Relevance to the cut-tire framework.} If the +conjecture holds, our cut-tire framework's domain assumption (= +cyclic edge connectivity exactly $6$ in $G^*$) is automatically +satisfied by every minimum $4$CT counterexample. If it doesn't +hold, we'd need to either prove that the counterexample is not +of the violating type, or extend the framework to higher-size +cuts. + +\end{document}