An extremal problem for vertex partition of complete multipartite graphs

Abstract

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