Packing tight Hamilton cycles in uniform hypergraphs

TL;DR

This paper proves that almost all edges of certain regular hypergraphs can be decomposed into specific types of Hamilton cycles, extending understanding of cycle decompositions in uniform hypergraphs.

Contribution

It introduces a new class of regular hypergraphs and demonstrates their edges can be nearly fully decomposed into type  Hamilton cycles, a novel result in hypergraph theory.

Findings

Almost all edges can be decomposed into Hamilton cycles

Applicable to a broad class of regular hypergraphs including random models

Extends cycle decomposition results to new hypergraph classes

Abstract

We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair of consecutive edges $E_{i-1},E_i$ in $C$ (in the natural ordering of the edges) we have $|E_{i-1}\setminus E_i|=\ell$. We define a class of $(\e,p)$-regular hypergraphs, that includes random hypergraphs, for which we can prove the existence of a decomposition of almost all edges into type $\ell$ Hamilton cycles, where $\ell<k/2$.