An extremal problem for vertex partition of complete multipartite graphs

For a graph GG and a family HH of graphs, a vertex partition of GG is called an HH-decomposition, if every part induces a graph isomorphic to one of HH. For 1≤a≤k1≤a≤k, let A(k,a)A(k,a) denote the graph which is a join of an empty graph of order aa and a complete graph of order k−ak−a. Let Ak={A(k,a):1≤a≤k}. In this paper, extremal problems related to HH-decomposition of a complete multipartite graph, where H⊂AkH⊂Ak, are studied. Among other results, it is proved that for every complete multipartite graph GG of order kℓkℓ, where ℓ≥k−2≥2ℓ≥k−2≥2, there is a positive integer aa such that GG admits an {A(k,a),A(k,a+1),A(k,a+2)}{A(k,a),A(k,a+1),A(k,a+2)}-decomposition.