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Local Convergence and Stability of Tight Bridge-addable Classes

  • G. Chapuy (a1) and G. Perarnau (a1)

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. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if ${\mathcal{G}}$ is bridge-addable and $G_{n}$ is a uniform $n$ -vertex graph from  ${\mathcal{G}}$ , then $G_{n}$ is connected with probability at least $(1+o_{n}(1))e^{-1/2}$ . The constant $e^{-1/2}$ is best possible, since it is reached for the class of all forests.

In this paper, we prove a form of uniqueness in this statement: if ${\mathcal{G}}$ is a bridge-addable class and the random graph $G_{n}$ is connected with probability close to  $e^{-1/2}$ , then $G_{n}$ is asymptotically close to a uniform $n$ -vertex random forest in a local sense. For example, if the probability converges to  $e^{-1/2}$ , then $G_{n}$ converges in the sense of Benjamini–Schramm to the uniformly infinite random forest  $F_{\infty }$ . This result is reminiscent of so-called “stability results” in extremal graph theory, the difference being that here the stable extremum is not a graph but a graph class.

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Supported by the European Research Council, grant ERC-2016-STG 716083 “CombiTop”.

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Local Convergence and Stability of Tight Bridge-addable Classes

  • G. Chapuy (a1) and G. Perarnau (a1)

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