The 334-Triangle Graph of $S{L}_3(\mathbb{Z})$

Long, Reid, and Thistlewaite have shown that some groups generated by representations of the $\mathrm{\Delta }334$ triangle group in $S{L}_3(\mathbb{Z})$ 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 $\mathrm{\Delta }334$ in the group. We provide examples of our graph for a variety of groups, and we use information about the graph for $S{L}_3(\mathbb{Z}\mathrm{/}2\mathbb{Z})$ to show that the chromatic number of the graph for $S{L}_3(\mathbb{Z})$ is at most eight. By generating a portion of the graph for $S{L}_3(\mathbb{Z})$ we show its chromatic number is at least four; we conjecture it is equal to four.