Hamiltonian cycles passing through matchings in $k$-ary $n$-cubes

TL;DR

This paper proves that in large enough $k$-ary $n$-cubes, any matching with up to $4n-20$ edges can be included in a Hamiltonian cycle, highlighting structural properties of these networks.

Contribution

It establishes a new bound for the inclusion of matchings in Hamiltonian cycles within $k$-ary $n$-cubes for specific parameters.

Findings

Any matching with at most $4n-20$ edges is contained in a Hamiltonian cycle for $n extgreater=5$, $k extgreater=4$.

The result extends understanding of Hamiltonian cycle structure in $k$-ary $n$-cubes.

Provides a bound on matchings that can be embedded in Hamiltonian cycles in these networks.

Abstract

As we all know, the $k$-ary $n$-cube is a highly efficient interconnect network topology structure. It is also a concept of great significance, with a broad range of applications spanning both mathematics and computer science. In this paper, we study the existence of Hamiltonian cycles passing through prescribed matchings in $k$-ary $n$-cubes, and obtain the following result. For $n\geq5$ and $k\geq4$, every matching with at most $4n-20$ edges is contained in a Hamiltonian cycle in the $k$-ary $n$-cube.