The competition numbers of complete multipartite graphs and orthogonal families of Latin squares

The competition graph of a digraphD is a graph which has the same vertex set as D and has an edge between u and v if and only if there exists a vertex x inD such that (u, x) and (v, x) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs to characterize a graph by its competition number. In this paper, we give new upper and lower bounds for the competition number of a complete multipartite graph Km n on m partite sets of the same size n by using orthogonal Latin squares. Furthermore, we give better bounds for the competition number of the complete tetrapartite graph K4 p for a prime number p.