Loose Hamilton paths in the 3-uniform cube hypergraph

TL;DR

This paper investigates the existence of loose Hamilton paths in the 3-uniform cube hypergraph, a hypergraph analogue of the hypercube, and identifies the dimensions in which such paths exist, given the impossibility of Hamilton cycles.

Contribution

It determines the dimensions where loose Hamilton paths exist in the 3-uniform cube hypergraph, extending understanding of Hamiltonian properties in hypergraph analogues of hypercubes.

Findings

Loose Hamilton paths exist in certain dimensions of the 3-uniform cube hypergraph.

The 3-uniform cube hypergraph cannot have a loose Hamilton cycle in any dimension.

The paper characterizes the dimensions supporting loose Hamilton paths.

Abstract

It is well-known that the $d$-dimensional hypercube contains a Hamilton cycle for $d\ge 2$. In this paper we address the analogous problem in the $3$-uniform cube hypergraph, a $3$-uniform analogue of the hypercube: for simple parity reasons, the $3$-uniform cube hypergraph can never admit a loose Hamilton cycle in any dimension, so we do the next best thing and consider loose Hamilton paths, and determine for which dimensions these exist.