Cycles and p-competition graphs

The notion of a p-competition graph and the p-competition number of a graph were introduced by S. -R. Kim, T. A. McKee, F. R. McMorris, and F. S. Roberts as a generalization of a competition graph and the competition number of a graph, respectively. Let p be a positive integer. The p-competition graph Cp(D) of a digraph D = (V, A) is a (simple undirected) graph which has the same vertex set V and has an edge between distinct vertices x and y if and only if there exist p distinct vertices v1, . . . , vp ∈ V such that (x, vi), (y, vi) are arcs of the digraph D for each i = 1, . . . , p. We can easily observe that, for any graph G, G together with sufficiently many isolated vertices is the p-competition number of some acyclic digraph. The p-competition number kp(G) of a graph G is defined to be the minimum number k such that G with k isolated vertices is the p-competition number of a acyclic digraph. In this paper, we give a necessary and sufficient condition for a cycle and for the complement of a cycle to be a p-competition graph. We also give a characterization of graphs whose pcompetition numbers are at most m, where m is a given nonnegative integer. Then, we compute the p-competition numbers of a cycle and the complement of a cycle.