One extra edge forces Berge pancyclicity

Abstract

We resolve a question of Bailey, Hollars, Li and Luo. For all sufficiently large $n$, let $r=\lfloor (n-1)/2\rfloor$. We prove that the edges of any Hamiltonian Berge cycle in a simple $n$-vertex $r$-uniform hypergraph, together with any one additional edge, contain Berge cycles of every length from $2$ to $n$. In odd order we prove a stronger prescribed-unused-edge theorem using rigidity of large subsets of odd cyclic groups and an alternating matching exchange. In even order we introduce a two-gap edge-reassignment method. Split locks cover all lengths outside a seven-term middle band. The absence of the central length forces an exact reflected translation-wave structure, which is eliminated by an additive covering theorem derived from sum-free stability. The remaining near-central lengths follow from a two-defect recurrence and bounded-run forcing.

48 pages, no figures. Substantially expanded version of v1: adds the even-order case and resolves the full Bailey-Hollars-Li-Luo one-extra-edge question. Title changed