A lower bound on the zero forcing number

Abstract

In this note, we study a dynamic vertex coloring for a graph $G$. In particular, one starts with a certain set of vertices black, and all other vertices white. Then, at each time step, a black vertex with exactly one white neighbor forces its white neighbor to become black. The initial set of black vertices is called a \emph{zero forcing set} if by iterating this process, all of the vertices in $G$ become black. The \emph{zero forcing number} of $G$ is the minimum cardinality of a zero forcing set in $G$, and is denoted by $Z(G)$. Davila and Kenter have conjectured in 2015 that $Z(G)\geq (g-3)(δ-2)+δ$ where $g$ and $δ$ denote the girth and the minimum degree of $G$, respectively. This conjecture has been proven for graphs with girth $g \leq 10$. In this note, we present a proof for $g \geq 5$, $δ\geq 2$, thereby settling the conjecture.