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 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 characterizing a graph by its competition number has been one of important research problems in the study of competition graphs. The Johnson graph J(n, d) has the vertex set {vX | X ∈ ([n] d ) }, where ([n] d ) denotes the set of all d-subsets of an n-set [n] = {1, . . . , n}, and two vertices vX1 and vX2 are adjacent if and only if |X1 ∩ X2| = d − 1. In this paper, we study the edge clique number and the competition number of J(n, d). Especially we give the exact competition numbers of J(n, 2) and J(n, 3).