The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs

Let RT (n), ED(n) and EOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, on n vertices. Then, as n →∞, RT (n) = (2n+1 πn )(n−1)/2 n1/2e−1/2 ( 1 + O(n−1/2+ ) ) (n odd), ED(n) = ( 4 πn )(n−1)/2 n1/2e−1/4 ( 1 + O(n−1/2+ ) ) , and EOG(n) = (3n+1 4πn )(n−1)/2 n1/2e−3/8 ( 1 + O(n−1/2+ ) ) , for any > 0. The last two families of graphs are also enumerated by their numbers of edges. The proofs use the saddle point method applied to appropriate n-dimensional integrals.