Characterizations of competition multigraphs

The notion of a competition multigraph was introduced by C. A. Anderson, K. F. Jones, J. R. Lundgren, and T. A. McKee [C. A. Anderson, K. F. Jones, J. R. Lundgren, and T. A. McKee: Competition multigraphs and the multicompetition number, Ars Combinatoria 29B (1990) 185–192] as a generalization of the competition graphs of digraphs.In this note, we give a characterization of competition multigraphs of arbitrary digraphs and a characterization of competition multigraphs of loopless digraphs. Moreover, we characterize multigraphs whose multicompetition numbers are at most mm, where mm is a given nonnegative integer and give characterizations of competition multihypergraphs.