Asymptotic enumeration of 2-covers and line graphs
Peter Cameron, Thomas Prellberg, Dudley Stark

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 -labelled vertices and for different types of related combinatorial objects called 2-covers. We find that the number of 2-covers, , and proper 2-covers, , on both have asymptotic growth where is the th Bell number, while the number of restricted 2-covers, , restricted, proper 2-covers on , , and line graphs , all have growth 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.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Stochastic processes and statistical mechanics · Advanced Combinatorial Mathematics
