Simplify level-cycle search table

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