On the girth of digraphs

It was conjectured by Caccetta and H aggkvist in 1978 that the girth of every digraph with n vertices and minimum outdegree r is at most dn=re. The conjecture was proved for r = 2 by Caccetta and H aggkvist, for r = 3 by Hamidoune and for r = 4; 5 by Ho ang and Reed. In this paper, the following two main results are proved: 1. The diameter of every strongly connected digraph of order n with girth g is at most n−g+ t, where t is the number of vertices having outdegree exactly 1. As a consequence, a short, self-contained proof of Caccetta and H aggkvist’s result is obtained. 2. The girth of every digraph with n vertices and minimum outdegree r is at most max{dn=re; 2r − 2}. As a consequence, the above conjecture is proved for the case n¿2r − 3r + 1. In other words, for each given r, the number of counterexamples to the conjecture, if any, is nite. c © 2000 Elsevier Science B.V. All rights reserved.