Asymptotic enumeration of 2-covers and line graphs

A 2-cover is a multiset of subsets of [n] := {1, 2, . . . , n} such that each element of [n] lies in exactly two of the subsets. A 2-cover is called proper if all of the subsets of distinct, and is called restricted if any two of them intersect in at most one element. In this paper we find asymptotic enumerations for the number of line graphs on n-labelled vertices and for 2-covers. We find that the number sn of 2-covers and the number tn of proper 2-covers both have asymptotic growth sn ∼ tn ∼ B2n2 exp ( − 2 log(2n/ log n) ) , where B2n is the 2nth Bell number. Moreover, the numbers un of restricted 2-covers on [n] and vn of restricted, proper 2-covers on [n] and ln of line graphs all have growth un ∼ vn ∼ ln ∼ B2n2n exp ( − [ 1 2 log(2n/ log n) ]2) . keywords: asymptotic enumeration, line graphs, set partitions