For a (finite and simple) graph $G=(V,E)$, the *vertex arboricity*, $va(G)$ of $G$ is defined to be the least integer $d$ such that the vertex set of $G$ has a partition $V=V_1\cup V_2\cup \ldots \cup V_d$ for which the induced subgraph $G[V_i]$ is a forest for each $i=1,2,\ldots,d$.

Question: Is it true that $\chi(G)\leq va(G)+2$ for any **triangle-free** graph $G$?

ps. The condition "triangle-freeness" of $G$ is necessary for the stated bound, since otherwise the complete graph $K_n$ for some $n\geq 6$ provides a counterexample!