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- W4300458488 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." @default.
- W4300458488 created "2022-10-03" @default.
- W4300458488 creator A5027585618 @default.
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- W4300458488 date "2015-04-23" @default.
- W4300458488 modified "2023-09-23" @default.
- W4300458488 title "Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture" @default.
- W4300458488 doi "https://doi.org/10.48550/arxiv.1504.06344" @default.
- W4300458488 hasPublicationYear "2015" @default.
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