  In this paper, we showcase the process of using an automated conjecturing
program called \emph{TxGraffiti} written and maintained by the second author.
We begin by proving a conjecture formulated by \emph{TxGraffiti} that for a
claw-free graph $G$, the vertex cover number $\beta(G)$ is greater than or
equal to the zero forcing number $Z(G)$. Our proof of this result is
constructive, and yields a polynomial time algorithm to find a zero forcing set
with cardinality $\beta(G)$. We also use the output of \emph{TxGraffiti} to
construct several infinite families of claw-free graphs for which
$Z(G)=\beta(G)$. Additionally, inspired by the aforementioned conjecture of
\emph{TxGraffiti}, we also prove a more general relation between the zero
forcing number and the vertex cover number for any connected graph with maximum
degree $\Delta \ge 3$, namely that $Z(G)\leq (\Delta-2)\beta(G)$+1.
