Partition graph into 2 or more claw-free subgraphs
It is NP-hard, even for $k=2$.

This is a special case of a theorem by Farrugia (Alastair Farrugia, Vertex-partitioning into fixed additive
induced-hereditary properties is NP-hard, The electronic journal of combinatorics 11 (2004), #R46), 
stating that it is NP-complete to test if the vertex set a graph can be partitioned into two subsets 
A and B such that G(A) belongs to the graph class P and G(B) belongs to the graph class Q, provided 
P and Q are closed under taking vertex-disjoint unions and talking induced subgraphs, and at least 
one of P and Q is non-trivial (meaning not all graphs in the class are edgeless).