Triangle-free Subgraphs at the Triangle-Free Process

TL;DR

This paper analyzes the number of specific triangle-free subgraphs in a random graph process, providing asymptotically tight estimates and concentration results for certain graph types at a critical stage.

Contribution

It introduces a new analysis combining branching processes and semi-random methods to estimate subgraph counts in the triangle-free process at a key point.

Findings

Asymptotically tight estimates for expected subgraph counts.

Concentration results for balanced graphs with density less than 2.

Analysis applies at the stage where about n^{3/2 + ε} edges are added.

Abstract

We consider the triangle-free process: given an integer n, start by taking a uniformly random ordering of the edges of the complete n-vertex graph K_n. Then, traverse the ordered edges and add each traversed edge to an (initially empty) evolving graph - unless its addition creates a triangle. We study the evolving graph at around the time where \Theta(n^{3/2 + \epsilon}) edges have been traversed for any fixed \epsilon \in (0,10^{-10}). At that time and for any fixed triangle-free graph F, we give an asymptotically tight estimation of the expected number of copies of F in the evolving graph. For F that is balanced and have density smaller than 2 (e.g., for F that is a cycle of length at least 4), our argument also gives a tight concentration result for the number of copies of F in the evolving graph. Our analysis combines Spencer's original branching process approach for analysing the triangle-free process and the semi-random method.