Constrained graph processes

Abstract

Let $\mathcal{Q}$ 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 $\mathcal{Q}$: start with $G$ the empty graph on $n$ vertices. Add edges to $G$ one at a time, each time choosing uniformly from all $e\in G^c$ such that $G+e\in \mathcal{Q}$. Stop when there are no such edges, so the graph $G_\infty$ reached is maximal in $\mathcal{Q}$. Erdős, Suen and Winkler asked how many edges the resulting graph typically has, giving good bounds for $\mathcal{Q}=\{$bipartite graphs$\}$ and $\mathcal{Q}=\{$triangle free graphs$\}$. We answer this question for $C_4$-free graphs and for $K_4$-free graphs, by considering a related question about standard random graphs $G_p\in \mathcal{G}(n,p)$. The main technique we use is the 'step by step' approach of [3]. We wish to show that $G_p$ has a certain property with high probability. For example, for $K_4$ free graphs the property is that every 'large' set $V$ of vertices contains a triangle not sharing an edge with any $K_4$ in $G_p$. We would like to apply a standard Martingale inequality, but the complicated dependence involved is not of the right form. Instead we examine $G_p$ 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].