Induced Subgraph Bounds on the Zero Forcing Number and a $(\chi, \omega, Z)$-Conjecture

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 $Z(G)\ge\delta(H)+1$. Motivated by this result and the bound $\chi(G) \leq Z(G) + 1 $ by Taklimi (2013), we conjecture that
\begin{equation*}
\chi(G) \leq \left \lceil \frac{\omega(G)+Z(G)+1}{2}\right\rceil. \end{equation*} As supporting evidence, we prove that the conjecture holds for triangle-free regular graphs and also provide numerical evidence.