Enumeration of Connected Uniform Hypergraphs

Tsiriniaina Andriamampianina (1) , Vlady Ravelomanana (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

In this paper, we are concerned in counting exactly and asymptotically connected labeled $b$-uniform hypergraphs ($b \geq 3$). Enumerative results on connected graphs are generalized here to connected uniform hypergraphs. For this purpose, these structures are counted according to the number of vertices and hyperedges. First, we show how to compute step by step the associated exponential generating functions (EGFs) by means of differential equations and provide combinatorial interpretations of the obtained results. Next, we turn on asymptotic enumeration. We establish Wright-like inequalities for hypergraphs and by means of complex analysis, we obtain the asymptotic number of connected $b$-uniform hypergraphs with $n$ vertices and $(n+\ell)/(b-1)$ hyperedges whenever $\ell = o(n^{1/3}/b^{1/3})$. This latter result confirms a conjecture made by Karo\'nski and {\L}uczak in \cite{KL97} about the validity of their formula for excesses in the `Wright's range'.

Mots clés: random hypergraphs; $b$-uniform hypergraphs; enumerative and analytic combinatorics; saddle-point method; generalized Wright's coefficients; random hypergraphs.

Domaines: Algorithme et structure de données [cs.DS]; Combinatoire [math.CO]

Tsiriniaina Andriamampianina, Vlady Ravelomanana. Enumeration of Connected Uniform Hypergraphs. 2005. ⟨hal-00004824⟩