Towards a hypergraph version of the P\'osa-Seymour conjecture

TL;DR

This paper advances hypergraph theory by establishing minimum codegree conditions that guarantee the existence of Hamilton cycle powers and certain spanning hypergraphs, moving towards a hypergraph analogue of the Pósa-Seymour conjecture.

Contribution

It proves new minimum codegree thresholds ensuring Hamilton cycle powers and spanning hypergraphs of bounded tree-width in hypergraphs, extending classical graph conjectures to hypergraphs.

Findings

Minimum codegree condition guarantees Hamilton cycle powers.

Thresholds also ensure spanning hypergraphs of bounded tree-width.

Results progress towards a hypergraph Pósa-Seymour conjecture.

Abstract

We prove that for fixed $r\ge k\ge 2$, every $k$-uniform hypergraph on $n$ vertices having minimum codegree at least $(1-(\binom{r-1}{k-1}+\binom{r-2}{k-2})^{-1})n+o(n)$ 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 P\'osa-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.