@inproceedings{Erds2015ApproximateCO,
title={Approximate Counting of Graphical Realizations},
author={P{\'e}ter L. Erdős and S{\'a}ndor Z. Kiss and Istv{\'a}n Mikl{\'o}s and Lajos Soukup and Arndt von Haeseler},
booktitle={PloS one},
year={2015}
}

In 1999 Kannan, Tetali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007), for regular directed graphs (by Greenhill, 2011) and for half-regular bipartite graphs (by Miklós, Erdős and Soukup, 2013). Several heuristics on counting the number of possible realizations exist (via… CONTINUE READING