Arc-disjoint hamiltonian paths in Cartesian products of directed cycles

TL;DR

This paper proves that the Cartesian product of two or more directed cycles (of length at least two) always contains two arc-disjoint Hamiltonian paths, answering a question from 1985 and extending to certain Cayley digraphs.

Contribution

It establishes the existence of two arc-disjoint Hamiltonian paths in Cartesian products of directed cycles, solving a long-standing open problem for two and four or more cycles.

Findings

Two arc-disjoint Hamiltonian paths exist in the Cartesian product of two directed cycles.

The same holds for the Cartesian product of four or more directed cycles.

Open cases remain for the product of three directed cycles.

Abstract

We show that if $C_1$ and $C_2$ are directed cycles (of length at least two), then the Cartesian product $C_1 \Box C_2$ has two arc-disjoint hamiltonian paths. (This answers a question asked by J. A. Gallian in 1985.) The same conclusion also holds for the Cartesian product of any four or more directed cycles (of length at least two), but some cases remain open for the Cartesian product of three directed cycles. We also discuss the existence of arc-disjoint hamiltonian paths in $2$-generated Cayley digraphs on (finite or infinite) abelian groups.