Connectivity for bridgeaddable monotone graph classes
Abstract
A class A of labelled graphs is bridgeaddable 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 bridgeaddable, monotone class A whose elements have vertex set 1,...,n, the probability that a uniformly random element of A is connected is at least (1o_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).
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 arXiv:
 arXiv:1110.0009
 Bibcode:
 2011arXiv1110.0009A
 Keywords:

 Mathematics  Combinatorics;
 60C05
 EPrint:
 11 pages