From ee4233f0668ddeb42b49ae0171067491e9977389 Mon Sep 17 00:00:00 2001 From: didericis Date: Wed, 27 May 2026 00:22:00 -0400 Subject: [PATCH] coloring_nested_tire_graphs: extend even-cycle note with 8-cut question MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Added section "Could the minimum non-trivial cyclic cut be 8?" Answer: yes in principle. Birkhoff gives "≥ 6"; nothing in the condition pins the value to 6. A planar cubic G^* with min non-facial cyclic edge connectivity = 8 would have: - No non-facial 6-edge cut. - No non-facial 7-edge cut. - Some non-facial 8-edge cut. By cut-parity lemma: 8-cuts have even sides; 7-cuts have odd sides. Heuristic when this happens: link vertices of degree ≥ 6 (rather than icosahedron-tight 5) push second-link length to ≥ 10, eliminating small non-facial separators. The cut-tire framework adapts: chain DP and T_∂ construction work for any cut size, with parameter changes. Per-tire half needs re-examining for larger structures. Bottom line: min non-trivial cyclic cut is one of {6, 7, 8, 9, ...}; Birkhoff doesn't pin it down. Framework's natural domain is whatever value it happens to take, with 6 being the simplest. Co-Authored-By: Claude Opus 4.7 --- .../notes/even_separating_cycle.aux | 6 ++- .../notes/even_separating_cycle.log | 10 ++--- .../notes/even_separating_cycle.pdf | Bin 169383 -> 174458 bytes .../notes/even_separating_cycle.tex | 42 ++++++++++++++++++ 4 files changed, 52 insertions(+), 6 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux index 4064ba6..606a1d8 100644 --- a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux +++ b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux @@ -7,4 +7,8 @@ \@writefile{toc}{\contentsline {paragraph}{However:}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Next layer.}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Relevance to the cut-tire framework.}{3}{}\protected@file@percent } -\gdef \@abspage@last{3} +\@writefile{toc}{\contentsline {paragraph}{Yes, in principle.}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{What would force this?}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{The framework adapts.}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Bottom line.}{4}{}\protected@file@percent } +\gdef \@abspage@last{4} diff --git a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.log b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.log index bd1a268..d1a8fea 100644 --- a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.log +++ b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.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 00:16 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:22 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -262,7 +262,7 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) [1 {/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] - + [4] (./even_separating_cycle.aux) ) Here is how much of TeX's memory you used: 3257 strings out of 478268 @@ -288,10 +288,10 @@ m/cmsy10.pfb> -Output written on even_separating_cycle.pdf (3 pages, 169383 bytes). +Output written on even_separating_cycle.pdf (4 pages, 174458 bytes). 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a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.tex +++ b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.tex @@ -185,4 +185,46 @@ hold, we'd need to either prove that the counterexample is not of the violating type, or extend the framework to higher-size cuts. +\section*{Could the minimum non-trivial cyclic cut be $8$?} + +\paragraph{Yes, in principle.} Birkhoff gives $\ge 6$; nothing in +the condition pins the value to $6$. A planar cubic $G^*$ with +non-facial cyclic edge connectivity \emph{exactly $8$} would have: +\begin{itemize} +\item No non-facial cyclic cut of size $6$ (= no separating + $6$-cycle in $G$ with $\ge 2$ vertices each side). +\item No non-facial cyclic cut of size $7$ (= no separating + $7$-cycle in $G$ with $\ge 2$ vertices each side). +\item Some non-facial cyclic cut of size $8$. +\end{itemize} + +By the cut-parity lemma, a size-$8$ cut would have even-sized +sides. Size-$7$ cuts would have odd-sized sides; for such cuts +to not exist non-facially, the graph would need a structural +parity barrier or just lack any odd-cardinality separations. + +\paragraph{What would force this?} Looking at the second-link +heuristic: if every vertex's link contains only vertices of degree +$\ge 6$ rather than the icosahedron-tight degree $5$, the +second-link length jumps to $\ge 5 + 5 \cdot 1 = 10$. Such graphs +exist (denser triangulations); whether such a graph is also a +minimum $4$CT counterexample (= class-2 cubic dual + planar + +internally $6$-connected) is unknown. + +\paragraph{The framework adapts.} Even if the minimum non-trivial +cyclic cut is $8$ (or some other value $> 6$), the cut-tire chain +DP doesn't structurally depend on cut size $= 6$. The same +constructions (cut tires, boundary cut tire $T_\partial$, chain DP +via shared edges) apply to $8$-edge cuts with minor parameter +changes. What \emph{does} change: per-tire enumeration size +scales with cut size, and the per-tire half (Prop 1.13) was proved +specifically for spoke-only cut tires with simple-cycle face +boundaries --- it would need re-examining for larger structures. + +\paragraph{Bottom line.} The minimum non-trivial cyclic cut size +for a hypothetical $4$CT counterexample is one of +$\{6, 7, 8, 9, \dots\}$, and Birkhoff alone doesn't pin it down. +The framework's natural domain is whichever value it happens to +take, with $6$ being the simplest case to enumerate and study. + \end{document}