  Long, Reid, and Thistlewaite have shown that some groups generated by
representations of the $\Delta 334$ triangle group in $SL_3({\mathbb Z})$ are
thin, while the status of others is unknown. In this paper we take a new
approach: for each group we introduce a new graph that captures information
about representations of $\Delta 334$ in the group. We provide examples of our
graph for a variety of groups, and we use information about the graph for
$SL_3({\mathbb Z}/2{\mathbb Z})$ to show that the chromatic number of the graph
for $SL_3({\mathbb Z})$ is at most eight. By generating a portion of the graph
for $SL_3({\mathbb Z})$ we show its chromatic number is at least four; we
conjecture it is equal to four.
