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Asymptotic enumeration of labelled graphs by genus
"... We obtain asymptotic formulas for the number of rooted 2connected and 3connected surface maps on an orientable surface of genus g with respect to vertices and edges simultaneously. We also derive the bivariate version of the large facewidth result for random 3connected maps. These results are the ..."
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We obtain asymptotic formulas for the number of rooted 2connected and 3connected surface maps on an orientable surface of genus g with respect to vertices and edges simultaneously. We also derive the bivariate version of the large facewidth result for random 3connected maps. These results are then used to derive asymptotic formulas for the number of labelled kconnected graphs of orientable genus g for k ≤ 3. 1
Logical limit laws for minorclosed classes of graphs. Submitted. Available from http://arxiv.org/abs/1401.7021
"... Let G be an addable, minorclosed class of graphs. We prove that the zeroone law holds in monadic secondorder logic (MSO) for the random graph drawn uniformly at random from all connected graphs in G on n vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs ..."
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Let G be an addable, minorclosed class of graphs. We prove that the zeroone law holds in monadic secondorder logic (MSO) for the random graph drawn uniformly at random from all connected graphs in G on n vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in G on n vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface S. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of length ≈ 5 · 10−6. Finally, we analyse examples of nonaddable classes where the behaviour is quite different. For instance, the zeroone law does not hold for the random caterpillar on n vertices, even in FO. 1