Proof of a conjecture on the zero forcing number of a graph

Abstract: Amos et al. (Discrete Appl. Math. 181 (2015) 1-10) introduced the notion of the $k$-forcing number of graph for a positive integer $k$ as the generalization of the zero forcing number of a graph. The $k$-forcing number of a simple graph $G$, denoted by $F_k(G)$, is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most $k$ non-colored neighbors, then each of its non-colored neighbors become colored. Particulary, $F_1(G)$ is a widely studied invariant with close connection to the maximum nullity of a graph, under the name of the zero forcing number, denoted by $Z(G)$. Among other things, the authors proved that for a connected graph $G$ of order $n$ with $\Delta=\Delta(G)\geq 2$, $Z(G)\leq \frac{(\Delta-2)n+2}{\Delta-1}$, and this inequality is sharp. Moreover, they conjectured that $Z(G)=\frac{(\Delta-2)n+2}{\Delta-1}$ if and only if $G=C_n$, $G=K_{\Delta+1}$ or $G=K_{\Delta, \Delta}$. In this note, we show the above conjecture is true.