Connectivity for Bridge-Addable Monotone Graph Classes

Abstract

A classof labelled graphs isbridge-addableif, for all graphsGinand all verticesuandvin distinct connected components ofG, the graph obtained by adding an edge betweenuandvis also in; the classismonotoneif, for allG∈and all subgraphsHofG, we haveH∈. We show that for any bridge-addable, monotone classwhose elements have vertex set {1,. . .,n}, the probability that a graph chosen uniformly at random fromis connected is at least (1−on(1))e−½, whereon(1) → 0 asn→ ∞. This establishes the special case of the conjecture of McDiarmid, Steger and Welsh when the condition of monotonicity is added. This result has also been obtained independently by Kang and Panagiotou.