Extremal digraphs avoiding distinct walks of length 3 with the same endpoints

TL;DR

This paper determines the maximum size of directed graphs on n vertices that avoid having two different walks of length 3 with the same start and end points, fully solving a problem posed in 2007.

Contribution

It characterizes the extremal digraphs avoiding repeated-length-3 walks with identical endpoints, completing a long-standing problem.

Findings

Maximum size of such digraphs is established

Extremal digraphs are characterized

Complete solution to Zhan's 2007 problem obtained

Abstract

In this paper, we determine the maximum size of digraphs on $n$ vertices in which there are no two distinct walks of length $3$ with the same initial vertex and the same terminal vertex. The digraphs attaining this maximum size are also characterized. Combining this with previous results, we obtain a full solution to a problem proposed by X. Zhan in 2007.