From Brooks' Theorem, we know that
if a graph $G$ satisfies that $\Delta (G)=4$ and there is no $5$-clique in $G$, then $\chi (G)\leq 4$.
And it is easy to find a counterexample to the following:
if a graph $G$ satisfies that $\Delta (G)=4$ and there is no $4$-clique in $G$, then $\chi (G)\leq 3$.
I want to ask whether the following conclusion is right or please give a counterexample!
if a graph $G$ satisfies that $\Delta (G)=4$ and there is no $3$-clique in $G$, then $\chi (G)\leq 3$.