Cycles in Oriented 3-graphs

TL;DR

This paper investigates the maximum possible girth of oriented 3-graphs and 3-tournaments, demonstrating asymptotically optimal bounds and contrasting behaviors with 2-tournaments.

Contribution

It establishes asymptotically tight bounds on the shortest cycle length in oriented 3-graphs and 3-tournaments, revealing contrasting properties with 2-tournaments.

Findings

Existence of oriented 3-graphs with girth proportional to n^2/2

Existence of 3-tournaments with girth proportional to n^2/3

Contrast between 3-graphs and 2-tournaments in cycle lengths

Abstract

An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph (on $n$ vertices) be? We show that there exist oriented 3-graphs whose shortest cycle has length $\frac{n^2}{2}(1+o(1))$: this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $\frac{n^2}{3}(1+o(1))$, in complete contrast to the case of 2-tournaments.