A Short Proof for a Lower Bound on the Zero Forcing Number

TL;DR

This paper presents a concise proof confirming a conjecture that establishes a lower bound on the zero forcing number of a graph based on its girth and minimum degree, advancing understanding in graph theory.

Contribution

The paper offers a short, elegant proof of a conjecture relating zero forcing number to girth and minimum degree in graphs, simplifying previous approaches.

Findings

Confirmed the conjecture for all graphs with girth at least 3 and minimum degree at least 2.

Established the lower bound Z(G) ≥ (g-2)(δ-2)+2.

Provided a more accessible proof technique for the conjecture.

Abstract

We provide a short proof of a conjecture of Davila and Kenter concerning a lower bound on the zero forcing number $Z(G)$ of a graph $G$. More specifically, we show that $Z(G)\geq (g-2)(\delta-2)+2$ for every graph $G$ of girth $g$ at least $3$ and minimum degree $\delta$ at least $2$.