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

In Wang et al. (2011), the authors proved that the 3-ary n-cube admits a Hamiltonian path between u and v   passing through every edges in a prescribed set PP with at most 2n-22n-2 edges if the subgraph induced by PP consists of pairwise vertex-disjoint paths, none of which has u or v as internal vertices or both of them as end-vertices. They proved the above results based on some Lemmas including Lemma 4.7 in the paper. However, the proof of Case 2.4 of Lemma 4.7 is incorrect. In this note, we will give a counterexample and provide a correct proof. Therefore, the main result in the above paper is still correct.