Vertex partition of a complete multipartite graph into two kinds of induced subgraphs

Let k , ℓ be positive integers. Let G be a graph of order k ℓ . It is shown that if G is a complete multipartite graph, G has a vertex partition V ( G ) = V 1 ∪ V 2 ∪ ⋯ ∪ V ℓ such that for some pair of graphs H 1 and H 2 of order k , the subgraph of G induced by V i is isomorphic to H 1 or H 2 for any i with 1 ≤ i ≤ ℓ . Furthermore, for graphs not necessarily complete multipartite, similar problems are discussed.