A Note on Extremal Digraphs Containing at Most t Walks of Length k 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? Denote $$f(t)=\max \{2t+1,2\left\lceil \sqrt{2t+9/4}+1/2\right\rceil +3\}$$ . In this paper, we prove that the maximum number is equal to $$n(n-1)/2$$ and the extremal digraph are the transitive tournaments when $$k\ge n-1\ge f(t)$$ . Based on this result, we may determine the maximum numbers and the extremal digraphs when $$k\ge f(t)$$ and n is sufficiently large, which generalizes the existing results. A conjecture is also presented.