Connectivity for bridge-addable monotone graph classes

Journal article

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).

Authors: Berry, L; McDiarmid, C; Reed, B

• Publication status: Published
• Journal: Combinatorics, Probability and Computing
• Volume: 21
• Issue: 6
• Pages: 803-815
• Publication date: 2011-09-30
• DOI: 10.1017/S0963548312000272
• EISSN: 1469-2163
• ISSN: 0963-5483
• Language: English
• Keywords: 60C05; math.CO