  Given a graph $\Gamma$ and a number $n$, the associated $n^{th}$ graph braid
group $B_n(\Gamma)$ is the fundamental group of the unordered configuration
space of $n$ points on $\Gamma$. \'{S}wi\k{a}tkowski showed that for a given
$\Gamma$ and $n$ large enough, there is a free abelian subgroup of
$B_n(\Gamma)$ of rank equal to the cohomological dimension of $B_n(\Gamma)$. In
this note, we observe that at the cost of possibly adding additional points, we
can find a subgroup of the same cohomological dimension which is a direct
product of non-abelian free groups, and give some applications.
