Short Cycles in Digraphs with Local Average Outdegree at Least Two

Suppose G is a strongly connected digraph with order n girth g and diameter d. We prove that d + g ≤ n if G contains no arcs (u, v) with deg(u) = 1 and deg(v) ≤ 2. Caccetta and Häggkvist showed in 1978 that any digraph of order n with minimum outdegree 2 contains a cycle of length at most dn/2e. Applying the abovementioned result, we improve their result by replacing the minimum outdegree condition by some weaker conditions involving the local average outdegree. In particular, we prove that, for any digraph G of order n, if either 1. G has minimum outdegree 1 and deg(u) + deg(v) ≥ 4 for all arcs (u, v), or 2. deg(u) + deg(v) ≥ 3 for all pairs of distinct vertices u, v, then G contains a cycle of length at most dn/2e.