Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, January 24, 2019, 12:15 pm

Duration: 30 minutes

Location: CAB G11

Speaker: Anita Liebenau (UNSW)

Asymptotic enumeration of graphs by degree sequence

We show that the distribution of the degree sequence of G(n,m) can be approximated by a sequence of n independent binomial variables Bin(n−1,p) for a large range of p. This covers the range left open by previous work. In fact, we prove asymptotic formulae for the number of graphs of a given degree sequence, which implies the result about the degree sequence of the random graph. These formulae were conjectured in 1990 and 1997. In particular, we provide an asymptotic formula for the number of d-regular graphs for all d. The enumeration results are obtained via a new method of degree switching and ``contraction'' mappings. These new methods are applicable in other settings such as bipartite graphs, digraphs and hypergraphs of a given degree sequence.

This is joint work Nick Wormald.

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