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arXiv · 1504.06344

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

Guillaume Chapuy·Guillem Perarnau
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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.

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Guillaume Chapuy, Guillem Perarnau. 2015-04-23. Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture. https://doi.org/10.1016/j.jctb.2018.09.004

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