Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structurewhich 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 $H^r(n,p)$.
@article{10_37236_10794,
author = {Oliver Cooley and Mihyun Kang and Julian Zalla},
title = {Loose cores and cycles in random hypergraphs},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {4},
doi = {10.37236/10794},
zbl = {1503.05108},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10794/}
}
TY - JOUR
AU - Oliver Cooley
AU - Mihyun Kang
AU - Julian Zalla
TI - Loose cores and cycles in random hypergraphs
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10794/
DO - 10.37236/10794
ID - 10_37236_10794
ER -
%0 Journal Article
%A Oliver Cooley
%A Mihyun Kang
%A Julian Zalla
%T Loose cores and cycles in random hypergraphs
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/10794/
%R 10.37236/10794
%F 10_37236_10794
Oliver Cooley; Mihyun Kang; Julian Zalla. Loose cores and cycles in random hypergraphs. The electronic journal of combinatorics, Tome 29 (2022) no. 4. doi: 10.37236/10794
