A note on short cycles in diagraphs

In 1977, Caccetta and Haggkvist [1] conjectured that if G is a directed graph with n vertices and if each vertex of G has outdegree at least k, then G contains a directed cycle of length at most In~k]. We shall refer to this conjecture as the C H conjecture. Trivially, this conjecture is true for k = 1, and it has been proved for k = 2 (Caccetta and Haggkvist [1]) and k = 3 (Hamildoune [3]). Chv~ital and Szemer6di [2] proved that the C H conjecture holds if we replace the bound on the length of the cycle by In~k] + 2500. In this paper, we prove that the C H conjecture holds for k ~< 5. Our main results are obtained by extending the arguments of Chv~ital and Szemer6di. In order to prove the C H conjecture for k ~< 5, we show first that if the conjecture fails for a small value of k, then it must fail on a reasonably small graph.