On a conjecture of TxGraffiti: Relating zero forcing and vertex covers in graphs

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