More on quasi-random graphs, subgraph counts and graph limits

TL;DR

This paper explores properties of graph sequences related to subgraph counts and their relation to quasi-randomness, using graph limit theory and algebraic methods to identify when these properties imply randomness.

Contribution

It introduces new quasi-random properties of graph sequences and analyzes cases where these properties hold or remain open, expanding understanding of graph limit behaviors.

Findings

Certain subgraph count properties are quasi-random in some cases.

Some cases remain open due to proof limitations.

The approach uses graph limits and algebraic translation of combinatorial problems.

Abstract

We study some properties of graphs (or, rather, graph sequences) defined by demanding that the number of subgraphs of a given type, with vertices in subsets of given sizes, approximatively equals the number expected in a random graph. It has been shown by several authors that several such conditions are quasi-random, but that there are exceptions. In order to understand this better, we investigate some new properties of this type. We show that these properties too are quasi-random, at least in some cases; however, there are also cases that are left as open problems, and we discuss why the proofs fail in these cases. The proofs are based on the theory of graph limits; and on the method and results developed by Janson (2011), this translates the combinatorial problem to an analytic problem, which then is translated to an algebraic problem.