AbstractIn this article we determine the maximum number of arcs of a strong diagraph of order n, without a cycle of length at least n−k, for n⩾k2+2k+5; thus we partially solve a conjecture of Bermond, Germa, Heydemann and Sotteau.We give a conjecture concerning the maximum number of arcs of a strong diagraph of order n, of minimum half-degree r, without a cycle of length at least n−k, for large n. We prove it for r=2,k=1 or k=2
