Bounds for the Zero-Forcing Number of Graphs with Large Girth

We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, $δ\le Z(G)$ where $δ$ is the minimum degree, in the triangle-free case. In particular, we show that $2 δ- 2 \le Z(G)$ 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 $δ$.

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