Tur\'an problems for digraphs avoiding distinct walks of a given length with the same endpoints

TL;DR

This paper determines the maximum number of edges in directed graphs of size n that avoid having two different walks of length k with the same start and end points, and characterizes the extremal graphs for k ≥ 5.

Contribution

It establishes the maximum size of such digraphs and characterizes the extremal structures for k ≥ 5, advancing Turán-type problems for directed graphs.

Findings

Maximum size of digraphs avoiding distinct walks of length k with same endpoints

Characterization of extremal digraphs for k ≥ 5

Extension of Turán problems to directed graphs with walk constraints

Abstract

Let $n \ge 5$ and $k\ge 4$ be positive integers. We determine the maximum size of digraphs of order n that avoid distinct walks of length k with the same endpoints. We also characterize the extremal digraphs attaining this maximum number when $k \ge 5$.