The $Q_2$-Free Process in the Hypercube

J Robert Johnson,Trevor Pinto

doi:10.37236/8864

Abstract

The generation of a random triangle-saturated graph via the triangle-free process has been studied extensively. In this short note our aim is to introduce an analogous process in the hypercube. Specifically, we consider the $Q_2$-free process in $Q_d$ and the random subgraph of $Q_d$ it generates. Our main result is that with high probability the graph resulting from this process has at least $cd^{2/3} 2^d$ edges. We also discuss a heuristic argument based on the differential equations method which suggests a stronger conjecture, and discuss the issues with making this rigorous. We conclude with some open questions related to this process.

Highlights

A graph G on vertex set V is F -saturated if it contains no copy of F as a subgraph but the addition of any new edge in V (2) \ E(G) creates a copy of F
For each i we have |E(Gi)| = i and Gi+1 is obtained from Gi by randomly adding a new edge chosen uniformly at random from all of the possible edges which do not create a copy of F
Using the differential equations method for random graph processes introduced by Rucinski and Wormald in [24], Bohman determined the order of M with high probability

Summary

Introduction

Let F be a (typically small) graph. A graph G on vertex set V is F -saturated if it contains no copy of F as a subgraph but the addition of any new edge in V (2) \ E(G) creates a copy of F. Using the differential equations method for random graph processes introduced by Rucinski and Wormald in [24] (see for instance [29] for a survey of the subject), Bohman determined the order of M with high probability. Our main result, proved, is that with high probability, the subgraph of Qd generated by the Q2-free process in Qd has at least cd2/32d edges, for some constant c. The problem of analysing this process may be a natural testing ground for extending the differential equations method further or developing new techniques Related to this heuristic, we describing how our process relates to an important general framework for constrained graph process introduced by Bennett and Bohman [4]. To our knowledge, this associated random process has not

Main Result

Heuristic

Further Questions