ASYMPTOTIC ENUMERATION OF k-EDGE-COLORED k-REGULAR GRAPHS

Let P(k, n) be the collection of all sets of k disjoint perfect matchings in a complete graph with 2n vertices. We prove that if k = p(n(5/6)), then vertical bar P(k, n)vertical bar similar to 1/k! ((2n)/2(n)n!)k ((2n)/(2n)(k)(2n - k)!)n . (1 - k/2n)(n/2)e(k/4) for n -> infinity. This improves upon an existing result of Bollobas [Combinatorics, London Math. Soc. Lecture Note Ser. 52, Cambridge University Press, Cambridge, New York, 1981, pp. 80-102] who solved this problem for constant k, and a more recent result of Lieby et al. [Combin. Probab. Comput., 18 (2009), pp. 533-549] where an estimate is obtained for k = o(n(1/3)).