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 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 ${\displaystyle \{v_X\mid X\in b\in om\left\{\left[n\right]\right\}\left\{d\right\}}$, where ${\displaystyle b\in om\left\{\left[n\right]\right\}\left\{d\right\}}$ denotes the set of all d-subsets of an n-set [n] = {1,..., n}, and two vertices ${\displaystyle v_{X₁}}$ and ${\displaystyle v_{X₂}}$ 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).