Paths of given length in tournaments

TL;DR

This paper establishes an upper bound on the number of walks of a given length in any tournament with n vertices, providing insight into the structure of such directed graphs.

Contribution

It proves a tight upper bound on the number of walks of length k in n-vertex tournaments, advancing understanding of their combinatorial properties.

Findings

Maximum walks of length k in tournaments is at most n((n-1)/2)^k

Bound is tight and applies to all n-vertex tournaments

Provides a new combinatorial limit for directed graph walks

Abstract

We prove that every $n$-vertex tournament has at most $n\left(\frac{n-1}{2}\right)^k$ walks of length $k$.