The Binding Number of a Digraph

摘要

Caccetta-Haeggkvist's Conjecture discusses the relation between the girth g(D) of a digraph D and the minimum outdegree δ~+(D) of D. The special case when g(D) = 3 has lately attracted wide attention. For an undirected graph G, the binding number tnnd(G)≥ 3/2 is a sufficient condition for G to have a triangle (cycle with length 3). In this paper we generalize the concept of binding numbers to digraphs and give some corresponding results. In particular, the value range of binding numbers is given, and the existence of digraphs with a given binding number is confirmed. By using the binding number of a digraph we give a condition that guarantees the existence of a directed triangle in the digraph. The relationship between binding number and connectivity is also discussed.