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with the level-cycle three-colour restriction with respect to $S$. \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. +$4 \leq n \leq 12$ vertices and searched the vertex sources for each graph. 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. +size of the search space and the number of triangulations that admit a +witness. \begin{table}[ht] \centering \small \setlength{\tabcolsep}{4pt} -\begin{tabular}{cccc} -$n$ & triangulations & with witness & source checks \\\hline -$4$ & $1$ & $1$ & $1$ \\ -$5$ & $1$ & $1$ & $1$ \\ -$6$ & $2$ & $2$ & $2$ \\ -$7$ & $5$ & $5$ & $5$ \\ -$8$ & $14$ & $14$ & $14$ \\ -$9$ & $50$ & $50$ & $50$ \\ -$10$ & $233$ & $233$ & $237$ \\ -$11$ & $1249$ & $1249$ & $1296$ \\ -$12$ & $7595$ & $7595$ & $8069$ \\ +\begin{tabular}{ccc} +$n$ & triangulations & with witness \\\hline +$4$ & $1$ & $1$ \\ +$5$ & $1$ & $1$ \\ +$6$ & $2$ & $2$ \\ +$7$ & $5$ & $5$ \\ +$8$ & $14$ & $14$ \\ +$9$ & $50$ & $50$ \\ +$10$ & $233$ & $233$ \\ +$11$ & $1249$ & $1249$ \\ +$12$ & $7595$ & $7595$ \\ \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}