Subgraphs of weakly quasi-random oriented graphs

TL;DR

This paper investigates the structure of oriented graphs that avoid certain subgraphs, establishing bounds on edge orientations and providing constructions that demonstrate optimality, thereby advancing understanding in quasi-random oriented graph theory.

Contribution

It introduces new bounds and constructions for $H$-free oriented graphs, particularly for four- and six-cycle subgraphs, and extends extremal results to dense graphs and general cases.

Findings

Bound on $e(A,B)$ is optimal up to a constant for four-cycle free graphs.

Established similar bounds for six-cycle free graphs.

Provided extremal results for dense graphs with arbitrary $H$.

Abstract

It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an active area of research, and is relatively well-understood in the theory of quasi-random graphs and extremal combinatorics. In this paper, we consider the simplest cases of such a general question for oriented graphs, and provide some results on the global behavior of the orientation of $D$. For the case that $H$ is an oriented four-cycle we prove: in every $H$-free oriented graph $D$, there is a pair $A,B\ssq V(D)$ such that $e(A,B)\ge e(D)^{2}/32|D|^{2}$ and $e(B,A)\le e(A,B)/2$. We give a random construction which shows that this bound on $e(A,B)$ is best possible (up to the constant). In addition, we prove a similar result for the case $H$ is an oriented six-cycle, and a more precise result in the case $D$ is dense and $H$ is arbitrary. We also consider the related extremal question in which no condition is put on the oriented graph $D$, and provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures.