On the strength of connectedness of a random hypergraph

TL;DR

This paper extends a classical connectivity result from random graphs to random hypergraphs, establishing the equivalence of minimal degree and k-connectedness thresholds in hypergraph processes.

Contribution

It generalizes Bollobás and Thomason's result to d-uniform hypergraphs, providing precise probability thresholds for k-connectedness.

Findings

Thresholds for k-connectedness in random hypergraphs are established.

Probability of k-connectedness converges to a specific exponential expression.

Results extend classical graph connectivity theory to hypergraphs.

Abstract

Bollob\'{a}s and Thomason (1985) proved that for each $k=k(n) \in [1, n-1]$, with high probability, the random graph process, where edges are added to vertex set $V=[n]$ uniformly at random one after another, is such that the stopping time of having minimal degree $k$ is equal to the stopping time of becoming $k$-(vertex-)connected. We extend this result to the $d$-uniform random hypergraph process, where $k$ and $d$ are fixed. Consequently, for $m=\frac{n}{d}(\ln n +(k-1)\ln \ln n +c)$ and $p=(d-1)! \frac{\ln n + (k-1) \ln \ln n +c}{n^{d-1}}$, the probability that the random hypergraph models $H_d(n, m)$ and $H_d(n, p)$ are $k$-connected tends to $e^{-e^{-c}/(k-1)!}.$