Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture

TL;DR

This paper proves a conjecture that in large bridge-addable graph classes, the probability of a random graph being connected approaches at least e^{-1/2}, using a novel local double counting method.

Contribution

It establishes the asymptotically optimal lower bound for connectivity probability in bridge-addable graph classes and introduces a new local double counting proof technique.

Findings

Asymptotic lower bound of e^{-1/2} for connectivity probability

Proof technique based on local double counting and multivariate optimization

Bound is tight, achieved by forests

Abstract

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 $\mathcal{G}_n$ 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.