Hamiltonian decompositions of the wreath product of hamiltonian decomposable digraphs

TL;DR

This paper proves that the wreath product of two hamiltonian decomposable directed graphs is also hamiltonian decomposable in most cases, advancing the understanding of Hamiltonian decompositions in graph theory.

Contribution

It confirms most cases of a conjecture on Hamiltonian decompositions of wreath products of directed graphs, with specific conditions on the graph orders and structures.

Findings

Wreath product of certain hamiltonian decomposable graphs is also hamiltonian decomposable.

Most open cases of the conjecture are affirmed.

Exceptions involve specific structures of the component graphs.

Abstract

We affirm most open cases of a conjecture that first appeared 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 a hamiltonian decomposable directed graph $G$, such that $|V(G)|$ is even and $|V(G)|\geqslant 2$, with a hamiltonian decomposable directed graph $H$, such that $|V(H)| \geqslant 4$, is also hamiltonian decomposable except possibly when $G$ is a directed cycle and $H$ is a directed graph of odd order that admits a decomposition into $c$ directed hamiltonian cycle where $c$ is odd and $3\leqslant c \leqslant |V(H)|-2$.