## Abstract

A class of graphs is \emph{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 $\mathcal{G}_n$ is any bridge-addable class of graphs on $n$ vertices, and $G_n$ is taken uniformly at random from $\mathcal{G}_n$, then $G_n$ is connected with probability at least $e^{-\frac{1}{2}} + o(1)$, when $n$ tends to infinity. This lower bound is asymptotically best possible since it is reached for forests.

Our proof uses a ``local double counting'' strategy that may be of independent interest, and 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 relative to a supermultiplicative functional.

Our proof uses a ``local double counting'' strategy that may be of independent interest, and 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 relative to a supermultiplicative functional.

Original language | English |
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Journal | Journal of Combinatorial Theory. Series B |

Early online date | 17 Sept 2018 |

DOIs | |

Publication status | E-pub ahead of print - 17 Sept 2018 |

## Keywords

- bridge-addable classes
- random graphs
- random forests
- connectivity