Weakly saturated random graphs

Abstract

As introduced by Bollobás, a graph is weakly -saturated if the complete graph is obtained by iteratively completing copies of minus an edge. For all graphs , we obtain an asymptotic lower bound for the critical threshold , at which point the Erdős–Rényi graph is likely to be weakly -saturated. We also prove an upper bound for , for all which are, in a sense, strictly balanced. In particular, we improve the upper bound by Balogh, Bollobás, and Morris for , and we conjecture that this is sharp up to constants.