From fdab1e2e33764ddd1fac2003d977732669abecdc Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 1 Jun 2026 13:33:29 -0400 Subject: [PATCH] Simplify level-cycle search table --- papers/coloring_nested_tire_graphs/paper.pdf | Bin 1000266 -> 1004075 bytes papers/coloring_nested_tire_graphs/paper.tex | 29 +++++++++++++++++++ 2 files changed, 29 insertions(+) diff --git a/papers/coloring_nested_tire_graphs/paper.pdf b/papers/coloring_nested_tire_graphs/paper.pdf index 138b8433056bc4d817930b579ca4cd325c066650..98924683e1c72c76059a7fac30d38a28ea7ba00c 100644 GIT binary patch delta 68318 zcmV(#K;*y5hCQpKJ%EG(gaU*Egam{Iga(8Mgb0KQvvKz{S&t*pV~scsgyYrkP3ea3)XEH6%~Dqim_uKfb$I+^0ZMvgLI8 zL=d;cE%x)b3y?1^7ku&k$?Nlzmv2SBuq=1VFU~I(LRw~>Tv(qopTpF8@$*vn(@f-= z=gYGn{`BTFQ{F6Je|xIL^6!6VKfFDA_uc6)=YKxoanbJcGe`Dnd2%i-%39Ekr#{uW1N2UPj2|T^5d!W%W{*g?p7Q8wf4*X zRa2JPs@~K$4bHgbs$U;|BxvOcJHlAzx$c%`lP$b_Yt)F=F_o*1^4))_5zBHPnH`e} z%QWwq{hlZ$3klAw=m9IdzNJS6O>N4L<%VWj6dDk(=og87ImJxJ 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a/papers/coloring_nested_tire_graphs/paper.tex b/papers/coloring_nested_tire_graphs/paper.tex index 3eed21f..6abc74f 100644 --- a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -1345,6 +1345,35 @@ $S \subseteq V(G)$ such that $G$ admits a proper $4$-vertex-colouring with the level-cycle three-colour restriction with respect to $S$. \end{conjecture} +\subsection*{Enumeration for small $n$} + +We exhaustively enumerated all plane triangulation isomorphism classes with +$4 \leq n \leq 12$ vertices and, for each graph, searched the vertex sources +in order until a witness colouring was found or every source had been tested. +No counterexample to Conjecture~\ref{conj:level-cycle-three-colour} appeared +in this range. Table~\ref{tab:level-cycle-three-colour-counts} records the +size of the search space and the amount of source-search work required. + +\begin{table}[ht] +\centering +\small +\setlength{\tabcolsep}{4pt} +\begin{tabular}{cccc} +$n$ & triangulations & with witness & source checks \\\hline +$4$ & $1$ & $1$ & $1$ \\ +$5$ & $1$ & $1$ & $1$ \\ +$6$ & $2$ & $2$ & $2$ \\ +$7$ & $5$ & $5$ & $5$ \\ +$8$ & $14$ & $14$ & $14$ \\ +$9$ & $50$ & $50$ & $50$ \\ +$10$ & $233$ & $233$ & $237$ \\ +$11$ & $1249$ & $1249$ & $1296$ \\ +$12$ & $7595$ & $7595$ & $8069$ \\ +\end{tabular} +\caption{Exhaustive vertex-source search for the level-cycle three-colour conjecture on all triangulation isomorphism classes with $4 \leq n \leq 12$. Every triangulation in this range admits at least one vertex source witnessing the conjecture.} +\label{tab:level-cycle-three-colour-counts} +\end{table} + \begin{definition}[Seam] \label{def:seam} A \emph{seam} of a maximal planar graph $G$ is a simple cycle