  In this paper, we study a dynamic coloring of the vertices of a graph $G$
that starts with an initial subset $S$ of colored vertices, with all remaining
vertices being non-colored. At each discrete time interval, a colored vertex
with exactly one non-colored neighbor forces this non-colored neighbor to be
colored. The initial set $S$ is called a forcing set of $G$ if, by iteratively
applying the forcing process, every vertex in $G$ becomes colored. The forcing
number, originally known as the \emph{zero forcing number}, and denoted $F(G)$,
of $G$ is the cardinality of a smallest forcing set of $G$. We study lower
bounds on the forcing number in terms of its minimum degree and girth, where
the girth $g$ of a graph is the length of a shortest cycle in the graph. Let
$G$ be a graph with minimum degree $\delta \ge 2$ and girth~$g \ge 3$. Davila
and Kenter [Theory and Applications of Graphs, Volume 2, Issue 2, Article 1,
2015] conjecture that $F(G) \ge \delta + (\delta-2)(g-3)$. This conjecture has
recently been proven for $g \le 6$. The conjecture is also proven when the
girth $g \ge 7$ and the minimum degree is sufficiently large. In particular, it
holds when $g = 7$ and $\delta \ge 481$, when $g = 8$ and $\delta \ge 649$,
when $g = 9$ and $\delta \ge 30$, and when $g = 10$ and $\delta \ge 34$. In
this paper, we prove the conjecture for $g \in \{7,8,9,10\}$ and for all values
of $\delta \ge 2$.
