Competitively orientable complete multipartite graphs

Abstract

We say that a digraph D is competitive if any pair of vertices has a common out-neighbor in D and that a graph G is competitively orientable if there exists a competitive orientation of G. The competition graph of a digraph D is defined as the graph with the vertex set V(D) and an edge uv if and only if u and v compete in D. The notion of competitive digraph arose while studying digraph whose competition graphs are complete. We derive some useful properties of competitively orientable graphs and show that a complete graph of order n is competitively orientable if and only if n≥7. Then we completely characterize a competitively orientable complete multipartite graph in terms of the sizes of its partite sets. Moreover, we present a way to build a competitive multipartite tournament in each of competitively orientable cases.