The Turán number of directed paths and oriented cycles

Brown et al. (J Combin Theory Ser B 15(1):77–93, 1973) considered Turán-type extremal problems for digraphs. However, to date there are very few results on this problem, even asymptotically. Let P_2,2 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 Turán number of P_2,2 , and considered it a more natural and interesting problem to determine the Turán number of directed cycles. Let P_k and C_k denote the directed path and the directed cycle of order k , respectively. In this paper we determine the maximum size of C_k -free digraphs of order n for all n,k ∈ℕ^* , as well as the extremal digraphs attaining this maximum size. Similar result is obtained for P_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_2,2 . In particular, for oriented even cycles, we classify which oriented even cycles inherit the difficulty of their underlying graphs and which do not.