The Turan number of directed paths and oriented cycles

Brown et al. (J Combin Theory Ser B 15(1):77-93, 1973) considered Turan-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let (P-2,P-2) over right arrow be the orientation of C-4 which consists of two 2-paths with the same initial and terminal vertices. Huang and Lyu [Discrete Math., 343 (5) (2020)] recently determined the Turan number of (P-2,P-2) over right arrow, and considered it amore natural and interesting problem to determine the Turan number of directed cycles. Let (P-k) over right arrow and (C-k) over right arrow denote the directed path and the directed cycle of order k, respectively. In this paper we determine the maximum size of (C-k) over right arrow -free digraphs of order n for all n, k is an element of N*, as well as the extremal digraphs attaining this maximum size. Similar result is obtained for (P-k) over right arrow k where n is large. In addition, we generalize the result of Huang and Lyu by characterizing the extremal digraphs avoiding an arbitrary orientation of C-4 except (P-2,P-2) over right arrow. In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not.