Lower Bounds for the Size of Random Maximal H-Free Graphs

TL;DR

This paper establishes new lower bounds on the expected size of random maximal H-free graphs, especially for regular, strictly 2-balanced graphs, improving previous bounds for Turán numbers of complete bipartite graphs.

Contribution

It provides novel lower bounds for the size of random maximal H-free graphs, advancing understanding of Turán numbers for bipartite graphs.

Findings

New lower bounds for the expected number of edges in M_n(H).

Improved bounds for Turán numbers of K_{r,r} for r > 4.

Enhanced understanding of the structure of maximal H-free graphs.

Abstract

We consider the next greedy randomized process for generating maximal H-free graphs: Given a fixed graph H and an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph. Then, construct an n-vertex graph, M_n(H), iteratively as follows. Traverse the permuted edges of the complete n-vertex graph and add each one to the (initially empty) evolving graph M_n(H) - unless its addition creates a copy of H. The result of this process is a maximal H-free graph M_n(H). The basic question we are concerned with in here is: What is the expected number of edges in M_n(H)? We give new lower bounds on the expected number of edges in M_n(H) for the case where H is a regular, strictly 2-balanced graph. In particular, we obtain new lower bounds for Turan numbers of complete balanced bipartite graphs K_{r,r}, for every fixed r > 4. This improves an old lower bound of Erdos and Spencer.