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In 1977, Caceetta 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 In~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.doi:10.1016/0012-365x(87)90122-1 fatcat:epokl7ceybca7njypkyyzemxnq