# On Hamilton Cycle Decompositions of Tensor Products of Graphs

## TL;DR

This paper proves that under certain conditions, the tensor product of two Hamilton cycle decomposable graphs, including some sparse circulant graphs, also admits a Hamilton cycle decomposition.

## Contribution

It extends previous conjectures by establishing Hamilton cycle decomposability of tensor products for specific classes of circulant and multigraphs.

## Key findings

- Tensor products of certain circulant graphs are Hamilton cycle decomposable.
- Proved conjecture for specific nonbipartite circulant graphs.
- Includes tensor products involving multigraphs.

## Abstract

A Hamiltonian decomposition of $G$ is a partition of its edge set into disjoint Hamilton cycles. Manikandan and Paulraja conjectured that if $G$ and $H$ are Hamilton cycle decomposable circulant graphs with at least one of them is nonbipartite, then their tensor product is Hamilton cycle decomposable. In this paper, we have proved that, if $G$ is a Hamilton cycle decomposable circulant graph with certain properties and $H$ is a Hamilton cycle decomposable multigraph, then their tensor product is Hamilton cycle decomposable. In particular, tensor products of certain sparse Hamilton cycle decomposable circulant graphs are Hamilton cycle decomposable.