Induced Subgraph Bounds on the Zero Forcing Number and Chromatic Consequences

Abstract

Let $G$ be a graph with chromatic number $\chi (G)$, clique number $\omega (G)$ 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 $\delta (H)\ge 3$, then $Z(G)\ge \delta (H)+1$. As consequences, we prove that every triangle-free graph satisfies $\chi (G)\le \max \{3,Z(G)\}$ and obtain an application to planar graphs. Moreover, we prove that $\chi (G)\le \frac{Z(G)}{2}+2$ for every triangle-free graph.