Bounds for the Zero-Forcing Number of Graphs with Large Girth

TL;DR

This paper establishes new bounds for the zero-forcing number in triangle-free graphs with large girth, improving previous bounds and applying results to the Graph Complement Conjecture for certain graph classes.

Contribution

It introduces improved lower bounds for the zero-forcing number in graphs with girth at least 5 and small cut sets, and applies these bounds to prove the Graph Complement Conjecture for a broad class of graphs.

Findings

Lower bound of 2δ - 2 for girth ≥ 5

Enhanced bounds when small cut sets are present

Proof of the Graph Complement Conjecture for specific graph classes

Abstract

We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, $\delta \le Z(G)$ where $\delta$ is the minimum degree, in the triangle-free case. In particular, we show that $2 \delta - 2 \le Z(G)$ for graphs with girth of at least 5, and this can be further improved when $G$ has a small cut set. Using these results, we are able to prove the Graph Complement Conjecture on minimum rank for a large class of graphs. Lastly, we make a conjecture that the lower bound for $Z(G)$ increases as a function of the girth, $g$, and $\delta$.