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Communication Dans Un Congrès Année : 2016

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

Résumé

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 class of bridge-addable 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. Previous results on this problem include the lower bound $e^{-1}+o(1)$ proved by McDiarmid, Steger and Welsh, and the successive improvements to $e^{-0.7983}+o(1)$ by Ballister, Bollob\'{a}s and Gerke, and to $e^{-2/3}+o(1)$ in an unpublished draft of Norin. The bound $e^{-\frac{1}{2}} + o(1)$ was already known in the special case of bridge-alterable classes, independently proved by Addario-Berry, McDiarmid, and Reed, and by Kang and Panagiotou. 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 weighted by a supermultiplicative functional.

Dates et versions

hal-01185320 , version 1 (19-08-2015)

Identifiants

Citer

Guillaume Chapuy, Guillem Perarnau. Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture. SODA 2016, Jan 2016, Arlington, United States. ⟨10.1137/1.9781611974331.ch107⟩. ⟨hal-01185320⟩
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