Towards a hypergraph version of the Pósa-Seymour conjecture

Abstract

We prove that for fixed r ≥ k ≥ 2 , every k-uniform hypergraph on n vertices having minimum codegree at least (1 − (r−1/ k−1) + (r−2 / k−2) −1) n + o(n) contains the (r − k + 1)th power of a tight Hamilton cycle. This result may be seen as a step towards a hypergraph version of the Pósa–Seymour conjecture. Moreover, we prove that the same bound on the codegree suffices for finding a copy of every spanning hypergraph of tree-width less than r which admits a tree decomposition where every vertex is in a bounded number of bags