A lower bound on the zero forcing number

TL;DR

This paper proves a conjecture relating the zero forcing number of a graph to its girth and minimum degree, establishing a lower bound for graphs with girth at least 5 and minimum degree at least 2.

Contribution

It provides a proof of Davila and Kenter's conjecture for graphs with girth ≥ 5 and minimum degree ≥ 2, confirming the lower bound on the zero forcing number.

Findings

Confirmed the conjecture for girth ≥ 5 and δ ≥ 2

Established a new lower bound on zero forcing number

Extended previous results to larger girth values

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 $Z(G)\geq (g-3)(\delta-2)+\delta$ where $g$ and $\delta$ 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 $g \geq 5$, $\delta \geq 2$, thereby settling the conjecture.