Loose cores and cycles in random hypergraphs

TL;DR

This paper introduces the concept of loose cores in hypergraphs, analyzes their phase transition behavior in random hypergraphs, and improves bounds on the longest loose cycle length.

Contribution

It defines loose cores in hypergraphs, characterizes their phase transition, and develops an algorithm to analyze their structure and size.

Findings

Loose cores undergo a phase transition at a critical threshold.

The asymptotic degree distribution of vertices in loose cores is determined.

An improved upper bound on the longest loose cycle length is established.

Abstract

Inspired by the study of loose cycles in hypergraphs, we define the \emph{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 $H^r(n,p)$.