Digraphs that contain at most t distinct walks of a given length with the same endpoints

Abstract

Let n, k, t be positive integers. What is the maximum number of arcs in a digraph on n vertices in which there are at most t distinct walks of length k with the same endpoints? Determine the extremal digraphs attaining the maximum number. When $$t=1$$ , the problem has been studied by Wu, by Huang and Zhan, by Huang, Lyu and Qiao, by Lyu in four papers, and they solved all the cases but $$k=3$$ . For $$t\ge 2$$ , Huang and Lyu proved that the maximum number is equal to $$n(n-1)/2$$ and the extremal digraph is the transitive tournament when $$n\ge 6t+2$$ and $$k\ge n-1$$ . They also discussed the maximum number for the case $$n=k+2,k+3,k+4$$ . In this paper, we solve the problem for the case $$k\ge 6t+1$$ and $$n\ge k+5$$ , and we also characterize the structures of the extremal digraphs for $$n=k+2,k+3,k+4$$ .