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arXiv · 0707.0664

Asymptotic enumeration of 2-covers and line graphs

Peter Cameron·Thomas Prellberg·Dudley Stark
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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.

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Peter Cameron, Thomas Prellberg, Dudley Stark. 2007-07-04. Asymptotic enumeration of 2-covers and line graphs. https://arxiv.org/abs/0707.0664

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