The cycle structure of a~random nonhomogeneous hypergraph on the subcritical stage of evolution

A. V. Shapovalov

Diskretnaya Matematika, Tome 19 (2007) no. 4, pp. 52-69

Résumé

We consider a random nonhomogeneous hypergraph on $n$ vertices with $M=M(n)$ edges, $M_i=M_i(n)$ edges consist of $i$ vertices, \begin{gather*} \lim_{n\to\infty}M_i/M=c_i,\quad c_i\ge0,\quad i=0,1,\dots,m,\\ c_0+c_1+\dots+c_m=1,\quad M=M_0+M_1+\dots+M_m. \end{gather*} For each edge, vertices are chosen by random and equiprobable sampling with replacement out of $n$ vertices. Under the condition that $n\to\infty$ and $$ 0\lim_{n\to\infty}\frac Mn\Biggl(\sum_{i=2}^mc_ii(i-1)\Biggr)^{-1} $$ we show that the probability that the random hypergraph consists of hypertrees and components with one cycle tends to one. Similar results for random graphs and random homogeneous hypergraphs have been obtained earlier.