  We investigate the zero-forcing number for triangle-free graphs. We improve
upon the trivial bound, $\delta \le Z(G)$ where $\delta$ is the minimum degree,
in the triangle-free case. In particular, we show that $2 \delta - 2 \le Z(G)$
for graphs with girth of at least 5, and this can be further improved when $G$
has a small cut set. Using these results, we are able to prove the Graph
Complement Conjecture on minimum rank for a large class of graphs. Lastly, we
make a conjecture that the lower bound for $Z(G)$ increases as a function of
the girth, $g$, and $\delta$.
