2024/10/29 by Teegan Bailey, Bailey, Teegan, Yupei Li +3 · 1 citation
Mathematics · #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2410.21733
A Berge cycle of length ℓ in a hypergraph is an alternating sequence of ℓ distinct vertices and ℓ distinct edges v1,e1,v2, …, v_ℓ, eℓ such that \vi, vi+1\ ⊆ ei for all i, with indices taken modulo ℓ. We call an n-vertex hypergraph pancyclic if it contains Berge cycles of every length from 3 to n. We prove a sharp Dirac-type result guaranteeing pancyclicity in uniform hypergraphs: for n ≥ 70, 3 ≤ r ≤ \lfloor (n-1)/2\rfloor - 2, if \cH is an n-vertex, r-uniform hypergraph with minimum degree at least \lfloor (n-1)/2 \rfloor \choose r-1 + 1, then \cH is pancyclic.