A stability theorem for Berge Hamiltonian cycles under a minimum degree condition

Yichen Wang; Dániel Gerbner; Xiamiao Zhao

paper

A stability theorem for Berge Hamiltonian cycles under a minimum degree condition

arXiv:2608.04603

Abstract

In this paper, we study extremal and stability problems for Berge Hamiltonian cycles in $r$ -uniform hypergraphs under a minimum degree condition. Let $g_{r} (n, t) = (r n - t) + t (r - 1 t)$ , and let $t = t (k)$ be the unique integer satisfying $(r - 1 t - 1) < k \leq (r - 1 t)$ . Using a sharp Pósa-type degree sequence theorem of Salia, we prove an extremal upper bound on the number of hyperedges in an $n$ -vertex $r$ -uniform hypergraph with minimum degree at least $k$ and with no Berge Hamiltonian cycle. We also prove a stability theorem in the dense range before the first minimizer of $g_{r} (n, t)$ : every near-extremal example is contained in one of two natural non-Hamiltonian constructions.