In 1977, Caccetta and Haggkvist conjectured that if G is a directed graph with n vertices and minimal outdegree k, then G contains a directed cycle of length at most [n/k]. This conjecture is known to be true for k ⩽ 3. In this paper, we present a proof of the conjecture for the cases k = 4 and k = 5. We also present a new conjecture which implies that of Caccetta and Haggkvist.
