Maximum matchings in sparse random graphs: Karp-Sipser revisited
Random Structures & Algorithms
The phase transition in a random hypergraph
Journal of Computational and Applied Mathematics - Special issue: Probabilistic methods in combinatorics and combinatorial optimization
Asymptotic enumeration of sparse graphs with a minimum degree constraint
Journal of Combinatorial Theory Series A
Counting connected graphs asymptotically
European Journal of Combinatorics - Special issue on extremal and probabilistic combinatorics
Another proof of Wright's inequalities
Information Processing Letters
Counting connected graphs and hypergraphs via the probabilistic method
Random Structures & Algorithms
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
The order of the giant component of random hypergraphs
Random Structures & Algorithms
Birth and growth of multicyclic components in random hypergraphs
Theoretical Computer Science
The diameter of sparse random graphs
Combinatorics, Probability and Computing
New graph polynomials from the bethe approximation of the ising partition function
Combinatorics, Probability and Computing
Asymptotic normality of the size of the giant component via a random walk
Journal of Combinatorial Theory Series B
The MAX-CUT of sparse random graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Limit theorems for random MAX-2-XORSAT
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Cores of random r-partite hypergraphs
Information Processing Letters
A practical minimal perfect hashing method
WEA'05 Proceedings of the 4th international conference on Experimental and Efficient Algorithms
Design strategies for minimal perfect hash functions
SAGA'07 Proceedings of the 4th international conference on Stochastic Algorithms: foundations and applications
Asymptotic normality of the size of the giant component in a random hypergraph
Random Structures & Algorithms
Anatomy of the giant component: The strictly supercritical regime
European Journal of Combinatorics
On z-analogue of Stepanov-Lomonosov-Polesskii inequality
Journal of Combinatorial Theory Series B
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The theme of this work is an "inside-out" approach to the enumeration of graphs. It is based on a well-known decomposition of a graph into its 2-core, i.e. the largest subgraph of minimum degree 2 or more, and a forest of trees attached. Using our earlier (asymptotic) formulae for the total number of 2-cores with a given number of vertices and edges, we solve the corresponding enumeration problem for the connected 2-cores. For a subrange of the parameters, we also enumerate those 2-cores by using a deeper inside-out notion of a kernel of a connected 2-core.Using this enumeration result in combination with Caley's formula for forests, we obtain an alternative and simpler proof of the asymptotic formula of Bender, Canfield and McKay for the number of connected graphs with n vertices and m edges, with improved error estimate for a range of m values.As another application, we study the limit joint distribution of three parameters of the giant component of a random graph with n vertices in the supercritical phase, when the difference between average vertex degree and 1 far exceeds n-1/3. The three parameters are defined in terms of the 2-core of the giant component, i.e. its largest subgraph of minimum degree 2 or more. They are the number of vertices in the 2-core, the excess (#edges - #vertices) of the 2-core, and the number of vertices not in the 2-core. We show that the limit distribution is jointly Gaussian throughout the whole supercritical phase. In particular, for the first time, the 2-core size is shown to be asymptotically normal, in the widest possible range of the average vertex degree.