Genus bounds in right-angled Artin groups

We show that in any right-angled Artin group whose defining graph has chromatic number $k$, every non-trivial element has stable commutator length at least $1\mathrm{/}(6k)$. Secondly, if the defining graph does not contain triangles, then every non-trivial element has stable commutator length at least $1\mathrm{/}20$. These results are obtained via an elementary geometric argument based on earlier work of Culler.