Loose cores and cycles in random hypergraphs

Abstract

Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structure which mirrors the close relationship between cycles and 2-cores in graphs. We prove that in the r-uniform binomial random hypergraph H r (n, p), the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes.Our main tool is an algorithm called CoreConstruct, which enables us to analyse a peeling process for the loose core. By analysing this algorithm we determine the asymptotic degree distribution of vertices in the loose core and in particular how many vertices and edges the loose core contains. As a corollary we obtain an improved upper bound on the length of the longest loose cycle in H r (n, p).

Paths, cycles and sprinkling in random hypergraphs

“…In the case when j = 1, the lower bound was Θ(ε 2 n) while the upper bound was Θ(εn). This upper bound in the case when j = 1 was subsequently improved by the author, Kang and Zalla [8] and shown to be Θ(ε 2 n) in the range when p = (1 + ε)p 0 (although the results of that paper also cover the range p = c/n for any constant c > 1). The strategy used was to prove an upper bound on the length of the longest loose cycle which transfers to an upper bound for loose paths using a standard sprinkling argument, just as has been often observed for graphs.…”