Decompositions of the wreath product of certain directed graphs into directed hamiltonian cycles

TL;DR

This paper proves that the wreath product of certain directed graphs with specific properties is hamiltonian decomposable, confirming parts of a longstanding conjecture, and also identifies cases where it is not decomposable.

Contribution

It establishes new conditions under which the wreath product of directed graphs is hamiltonian decomposable, advancing understanding of graph decompositions.

Findings

Wreath product of certain hamiltonian decomposable graphs is hamiltonian decomposable.

Identifies cases where the wreath product is not hamiltonian decomposable.

Confirms special cases of a conjecture from 1987.

Abstract

We affirm several special cases of a conjecture that first appears in Alspach et al.~(1987) which stipulates that the wreath (lexicographic) product of two hamiltonian decomposable directed graphs is also hamiltonian decomposable. Specifically, we show that the wreath product of hamiltonian decomposable directed graph $G$, such that $|V(G)|$ is even and $|V(G)|\geqslant 3$, with a directed $m$-cycle such that $m \geqslant 4$ or the complete symmetric directed graph on $m$ vertices such that $m\geqslant 3$, is hamiltonian decomposable. We also show the wreath product of a directed $n$-cycle, where $n$ is even, with a directed $m$-cycle, where $m \in \{2,3\}$, is not hamiltonian decomposable.