Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture
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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 is any bridge-addable class of graphs on n vertices, and is taken uniformly at random from , then is connected with probability at least , 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.
CitationChapuy, G.; Perarnau-Llobet, G. Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture. "Journal of combinatorial theory. Series B", 17 Setembre 2018, vol. 136, p. 44-71.