A Note on Even Cycles and Quasi-Random Tournaments

TL;DR

This paper proves that if a tournament has nearly half of its k-cycles even for fixed even k ≥ 4, then the tournament is quasi-random, resolving a 1991 open question by Chung and Graham.

Contribution

It establishes a new characterization of quasi-random tournaments based on the parity of cycles, answering an open problem from 1991.

Findings

Nearly half of the k-cycles are even in quasi-random tournaments

The parity condition characterizes quasi-randomness for fixed even k ≥ 4

Resolves a long-standing open question in tournament theory

Abstract

A cycle C={v_1,v_2,....,v_1} in a tournament T is said to be even, if when walking along C, an even number of edges point in the wrong direction, that is, they are directed from v_{i+1} to v_i. In this short paper, we show that for every fixed even integer k >= 4, if close to half of the k-cycles in a tournament T are even, then T must be quasi-random. This resolves an open question raised in 1991 by Chung and Graham