Connectivity for bridge-addable monotone graph classes

TL;DR

This paper proves that for large graphs in a certain class with specific connectivity properties, the probability of being connected approaches e^{-1/2}, confirming a conjecture under added conditions.

Contribution

It establishes the probability bound for connectedness in bridge-addable, monotone graph classes, extending previous conjectures with new conditions.

Findings

Probability of connectivity approaches e^{-1/2} as n grows large.

Confirms a special case of a conjecture by McDiarmid, Steger, and Welsh.

Independent proof by Kang and Panagiotiou (2011).

Abstract

A class A of labelled graphs is bridge-addable if for all graphs G in A and all vertices u and v in distinct connected components of G, the graph obtained by adding an edge between u and u is also in A; the class A is monotone if for all G in A and all subgraphs H of G, H is also in A. We show that for any bridge-addable, monotone class A whose elements have vertex set 1,...,n, the probability that a uniformly random element of A is connected is at least (1-o_n(1)) e^{-1/2}, where o_n(1) tends to zero as n tends to infinity. This establishes the special case of a conjecture of McDiarmid, Steger and Welsh when the condition of monotonicity is added. This result has also been obtained independently by Kang and Panagiotiou (2011).