Constrained Graph Processes

Let Q be a monotone decreasing property of graphs G on n vertices. Erdős, Suen and Winkler [5] introduced the following natural way of choosing a random maximal graph in Q: start with G the empty graph on n vertices. Add edges to G one at a time, each time choosing uniformly from all e ∈ G such that G + e ∈ Q. Stop when there are no such edges, so the graph G∞ reached is maximal in Q. Erdős, Suen and Winkler asked how many edges the resulting graph typically has, giving good bounds for Q = {bipartite graphs} and Q = {triangle free graphs}. We answer this question for C4-free graphs and for K4-free graphs, by considering a related question about standard random graphs Gp ∈ G(n, p). The main technique we use is the ‘step by step’ approach of [3]. We wish to show that Gp has a certain property with high probability. For example, for K4 free graphs the property is that every ‘large’ set V of vertices contains a triangle not sharing an edge with any K4 in Gp. We would like to apply a standard Martingale inequality, but the complicated dependence involved is not of the right form. Instead we examine Gp one step at a time in such a way that the dependence on what has gone before can be split into ‘positive’ and ‘negative’ parts, using the notions of up-sets and down-sets. The relatively simple positive part is then estimated directly. The much more complicated negative part can simply be ignored, as shown in [3].