Asymptotic enumeration of 2-covers and line graphs

TL;DR

This paper derives asymptotic formulas for counting line graphs and various types of 2-covers on labeled vertices, revealing their growth rates and using probabilistic and generating function techniques.

Contribution

It provides the first asymptotic enumeration formulas for 2-covers, proper 2-covers, restricted 2-covers, and line graphs, with novel probabilistic and generating function methods.

Findings

Number of 2-covers and proper 2-covers grow as B_{2n}2^{-n}√(log n / 2n)

Number of restricted 2-covers, restricted proper 2-covers, and line graphs grow as B_{2n}2^{-n}n^{-1/2}exp(-[½log(2n/log n)]^2)

Different enumeration techniques are used for unrestricted and restricted types, including probabilistic and generating function methods.

Abstract

In this paper we find asymptotic enumerations for the number of line graphs on $n$-labelled vertices and for different types of related combinatorial objects called 2-covers. We find that the number of 2-covers, $s_n$, and proper 2-covers, $t_n$, on $[n]$ both have asymptotic growth $$ s_n\sim t_n\sim B_{2n}2^{-n}\exp(-\frac12\log(2n/\log n))= B_{2n}2^{-n}\sqrt{\frac{\log n}{2n}}, $$ where $B_{2n}$ is the $2n$th Bell number, while the number of restricted 2-covers, $u_n$, restricted, proper 2-covers on $[n]$, $v_n$, and line graphs $l_n$, all have growth $$ u_n\sim v_n\sim l_n\sim B_{2n}2^{-n}n^{-1/2}\exp(-[\frac12\log(2n/\log n)]^2). $$ In our proofs we use probabilistic arguments for the unrestricted types of 2-covers and and generating function methods for the restricted types of 2-covers and line graphs.