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 are distinct, and is called restricted if any two of them intersect in at most one element. s paper we find asymptotic enumerations for the number of line graphs on n labelled vertices and for 2-covers. d that the number s n of 2-covers and the number t n of proper 2-covers both have asymptotic growth s n ∼ t n ∼ B 2 n 2 − n exp ( − 1 2 log ( 2 n / log n ) ) , where B 2 n is the 2 n th Bell number. Moreover, the numbers u n of restricted 2-covers on [ n ] and v n of restricted, proper 2-covers on [ n ] and l n of line graphs all have growth u n ∼ v n ∼ l n ∼ B 2 n 2 − n n − 1 / 2 exp ( − [ 1 2 log ( 2 n / log n ) ] 2 ) .