Transversals and competition numbers of complete multipartite graphs

Let GG denote a simple graph and let IkIk denote the graph on kk isolated vertices. The competition number of GG is the minimum integer kk such that G∪IkG∪Ik is the competition graph of an acyclic digraph. We show that if n≥5n≥5 is odd, then the competition number of the complete tetrapartite graph Kn4 is at most n2−4n+8n2−4n+8. We also show that if nn is a prime integer and m≤nm≤n, then the competition number of Knm is at most n2−2n+3n2−2n+3.