This is a question inspired by the H-free cut problem. Given a graph, a partition of its vertex set $V$ into $r$ parts $V_1, V_2, \ldots, V_r$ is $H$-free if $G[V_i]$ does not induce a copy of $H$ for all $i$, $1 \leq i \leq r$.
I wish to consider the following question:
What is the least $r$ for which there exists a $H$-free partition into $r$ parts?
Notice that when $H$ is a single edge, then this amounts to finding the chromatic number, and is already NP-complete. I am wondering if it is easier to show NP-completeness for any fixed $H$ for this problem (easier, as compared to showing it for $H$-free cut). I even thought it might be obvious, but I didn't get anywhere. It is entirely possible I'm missing something quite straightforward, and if this is the case, I'd appreciate some pointers!