Induced Subgraph Bounds on the Zero Forcing Number and Chromatic Consequences

Abstract

Let $G$ be a graph with chromatic number $χ(G)$, clique number $ω(G)$ and zero forcing number $Z(G)$. We establish new lower bounds on $Z(G)$ in terms of induced triangle-free subgraphs. In particular, we show that if a graph $G$ contains an induced triangle-free subgraph $H$ with minimum degree $δ(H) \ge 3$, then $Z(G)\geδ(H)+1$. As consequences, we prove that every triangle-free graph satisfies $χ(G)\le\max\{3,Z(G)\}$ and obtain an application to planar graphs. Moreover, we prove that $χ(G)\le \frac{Z(G)}{2}+2$ for every triangle-free graph.