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

Let kk, ℓℓ be positive integers. Let GG be a graph of order kℓkℓ. It is shown that if GG is a complete multipartite graph, GG has a vertex partition V(G)=V1∪V2∪⋯∪VℓV(G)=V1∪V2∪⋯∪Vℓ such that for some pair of graphs H1H1 and H2H2 of order kk, the subgraph of GG induced by ViVi is isomorphic to H1H1 or H2H2 for any ii with 1≤i≤ℓ1≤i≤ℓ. Furthermore, for graphs not necessarily complete multipartite, similar problems are discussed.