Connectivity in graph classes

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• Guillem Perarnau Llobet (Birmingham)
• Thursday 29 October 2015, 14:30-15:30
• MR12.

If you have a question about this talk, please contact Andrew Thomason.

A class of graphs is bridge-addable if, given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. We prove a conjecture of McDiarmid, Steger and Welsh, that says that if G_n is taken uniformly at random from a class of bridge-addable graphs on n vertices, then G_n is connected with probability at least exp(-1/2)+o(1), when n tends to infinity. This lower bound is asymptotically best possible and it is reached for the class of forests. Our proof uses a “local double counting” strategy that enables us to compare the size of two sets of combinatorial objects by solving a related multivariate optimization problem. In our case, the optimization problem deals with partition functions of trees weighted by a supermultiplicative functional. This is joint work with Guillaume Chapuy.

This talk is part of the Combinatorics Seminar series.

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