TY - JOUR
AB - 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 $H^r(n,p)$.
AU - Cooley, Oliver
AU - Kang, Mihyun
AU - Zalla, Julian
ID - 12286
IS - 4
JF - The Electronic Journal of Combinatorics
KW - Computational Theory and Mathematics
KW - Geometry and Topology
KW - Theoretical Computer Science
KW - Applied Mathematics
KW - Discrete Mathematics and Combinatorics
TI - Loose cores and cycles in random hypergraphs
VL - 29
ER -