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

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 $n\ge 5$ and $k\ge 4$, every matching with at most $4n-20$ edges is contained in a Hamiltonian cycle in the $k$-ary $n$-cube.