Asymptotic enumeration of sparse graphs with a minimum degree constraint

@article{Pittel2003AsymptoticEO,
title={Asymptotic enumeration of sparse graphs with a minimum degree constraint},
author={Boris Pittel and Nicholas C. Wormald},
journal={J. Comb. Theory, Ser. A},
year={2003},
volume={101},
pages={249-263}
}

We derive an asymptotic formula for the number of graphs with n vertices all of degree at least k, and m edges, with k fixed. This is done by summing the asymptotic formula for the number of graphs with a given degree sequence, all degrees at least k. This approach requires analysis of a set of independent truncated Poisson variables, which approximate the degree sequence of a random graph chosen uniformly at random among all graphs with n vertices, m edges, and a minimum degree at least k. Ourâ€¦Â CONTINUE READING