Almost Symmetric Cycles in Large Digraphs

We prove that every digraph D with n≥7, n≥[inline-graphic not available: see fulltext]+6 vertices and at least (n−k−1)(n−1)+k(k+1) arcs contains all symmetric cycles of length at most n−k−2, an almost symmetric cycle of length n−k−1, and with some exceptions, also an almost symmetric cycle of length n−k. Consequently, D contains all orientations of cycles of length at most n−k, unless D is an exception.