Cycle decompositions in $k$-uniform hypergraphs

TL;DR

This paper proves that dense $k$-uniform hypergraphs can be decomposed into tight cycles and contain tight Euler tours, introducing new conditions for hypergraph decompositions and absorber constructions.

Contribution

It establishes new decomposition results for dense hypergraphs into tight cycles and provides an alternative absorber construction method.

Findings

Hypergraphs with high codegree decompose into tight cycles

Existence of tight Euler tours in such hypergraphs

New absorber construction conditions for hypergraph decompositions

Abstract

We show that $k$-uniform hypergraphs on $n$ vertices whose codegree is at least $(2/3 + o(1))n$ can be decomposed into tight cycles, subject to the trivial divisibility conditions. As a corollary, we show those graphs contain tight Euler tours as well. In passing, we also investigate decompositions into tight paths. In addition, we also prove an alternative condition for building absorbers for edge-decompositions of arbitrary $k$-uniform hypergraphs, which should be of independent interest.