Quasi-random oriented graphs

TL;DR

This paper establishes the equivalence of several quasi-randomness conditions in oriented graphs, extending known results from unoriented graphs and tournaments to more general underlying graphs.

Contribution

It generalizes the main results of Chung and Graham to arbitrary underlying graphs G, showing equivalences of quasi-randomness conditions in oriented graphs.

Findings

Conditions on oriented graphs are equivalent with high probability.

Exactly two orientations of a four-cycle are sufficient for quasi-randomness.

Results extend to general underlying graphs beyond complete graphs.

Abstract

We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham and Wilson in the case of unoriented graphs, and by Chung and Graham in the case of tournaments. Indeed, our main theorem extends to the case of a general underlying graph G the main result of Chung and Graham which corresponds to the case that G is complete. One interesting aspect of these results is that exactly two of the four orientations of a four-cycle can be used for a quasi-randomness condition, i.e., if the number of appearances they make in D is close to the expected number in a random orientation of the same underlying graph, then the same is true for every small oriented graph H