The competition numbers of Johnson graphs

The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x; v) and (y; v) are arcs of D. For any graph G, G together with sucien tly many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is dened 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 to characterize all graphs with given competition number k has been one of the important research problems in the study of competition graphs. The Johnson graph J(n; d) has the vertex set fvX j X 2 [n] d g, where [n]