Asymptotic enumeration of sparse graphs with a minimum degree constraint

Authors:
Boris Pittel;Nicholas C. Wormald
Affiliations:
Department of Mathematics, Ohio State University, Columbus, OH;Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada N2L 3G1 and Department of Mathematics and Statistics, University of Melbourne VIC 3010, Australia
Venue:
Journal of Combinatorial Theory Series A
Year:
2003

Citing 4
Cited 4

A graph-valued Markov process as rings-allowed polymerization model: subcritical behavior

SIAM Journal on Applied Mathematics
Sudden emergence of a giant k-core in a random graph

Journal of Combinatorial Theory Series B
The asymptotic number of labeled graphs with n vertices, q edges, and no isolated vertices

Journal of Combinatorial Theory Series A
Maximum matchings in sparse random graphs: Karp-Sipser revisited

Random Structures & Algorithms

Counting connected graphs inside-out

Journal of Combinatorial Theory Series B
Corrigendum: Corrigendum to “Counting connected graphs inside-out” [J. Combin. Theory Ser. B 93 (2005) 127--172]

Journal of Combinatorial Theory Series B
Local Limit Theorems for the Giant Component of Random Hypergraphs

APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Cores of random r-partite hypergraphs

Information Processing Letters

Quantified Score

Hi-index	0.00

Visualization

Abstract

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. Our main result generalizes a result of Bender, Canfield and McKay and of Korshunov, who treated the case k = 1 using different methods.