The zero forcing number of claw-free cubic graphs

Abstract

Let $G$ be a simple graph of order $n$. Let $S$ be a coloring subset of $V\left(G\right)$. The forcing process is that a colored vertex forces the uncolored neighbor to be colored if it has exactly one uncolored neighbor. The set $S$ is a zero forcing set if all vertices of $G$ become colored by iteratively applying the forcing process. The minimum size of a zero forcing set in a graph $G$ is zero forcing number, denoted by $Z\left(G\right)$, which is proposed in 2008 as a natural upper bound of the maximum nullity regarding the graph $G$. In the paper, we bound the zero forcing number in connected claw-free cubic graphs. More exactly if $G(\ne K_4)$ is a connected claw-free cubic graph with order $n$, then we prove that $Z\left(G\right)\le \alpha \left(G\right)$ except for three graphs with small order, and then $Z\left(G\right)\le \frac{n}{4}+1$ except for three classes of graphs. In fact, our results give affirmative answers for two open problems raised by Davila and Henning.