Subgraphs of Weakly Quasi-Random Oriented Graphs

• Authors: Omid Amini;Simon Griffiths;Florian Huc
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2011

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 where $H$ is an oriented four-cycle we prove the following: in every $H$-free oriented graph $D$, there is a pair $A,B\subseteq V(D)$ such that $e(A,B)\geq e(D)^{2}/32|D|^{2}$ and $e(B,A)\leq e(A,B)/2$. 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 where $H$ is an oriented six-cycle and a more precise result in the case where $D$ 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 we provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures.