Abstract
We consider the following definition of connectedness in $k$-uniform hypergraphs: two $j$-sets (sets of $j$ vertices) are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. The hypergraph is $j$-connected if all $j$-sets are pairwise $j$-connected. We determine the threshold at which the random $k$-uniform hypergraph with edge probability $p$ becomes $j$-connected with high probability. We also deduce a hitting time result for the random hypergraph process – the hypergraph becomes $j$-connected at exactly the moment when the last isolated $j$-set disappears. This generalises the classical hitting time result of Bollobás and Thomason for graphs.
Highlights
1.1 Preliminaries and main resultsIn the study of random graphs, one of the most well-known results concerns the hitting time for connectedness
We consider the following definition of connectedness in k-uniform hypergraphs: two j-sets are j-connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices
We determine the threshold at which the random k-uniform hypergraph with edge probability p becomes j-connected with high probability
Summary
In the study of random graphs, one of the most well-known results concerns the hitting time for connectedness. If we add randomly chosen edges one by one to an initially empty graph on n vertices, with high probability at the moment the last isolated vertex gains its first edge, the whole graph will become connected (this classical result was first proved by Bollobas and Thomason in [3]). The uniform model and the associated hypergraph process allow us to formulate exact hitting time results such as Theorem 1. The drawback is that the analysis of the model can become tricky due to the fact that the presence of different edges is not independent (the total number is fixed) For this reason, it is often easier to analyse the binomial model : Hk(n, p) is a random k-uniform hypergraph on vertex set {1, .
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