Hamiltonian paths and cycles with prescribed edges in the 3-ary n-cube

The k-ary n  -cube has been one of the most popular interconnection networks for massively parallel systems. Given a set PP of at most 2n − 2 (n ⩾ 2) prescribed edges and two vertices u and v, we show that the 3-ary n-cube contains a Hamiltonian path between u and v   passing through all edges of PP if and only if the subgraph induced by PP consists of pairwise vertex-disjoint paths, none of them having u or v as internal vertices or both of them as end-vertices. As an immediate result, the 3-ary n  -cube contains a Hamiltonian cycle passing through a set PP of at most 2n − 1 prescribed edges if and only if the subgraph induced by PP consists of pairwise vertex-disjoint paths.