Embedding Cycles into Hypercubes with Prescribe Vertices in the Specific Order

In this paper, we are interesting in a new cycle embedding problem. Let x₁, x₂,..., x_k be any k-vertices. Can we find a cycle C in the hypercube Q_n such that C traverses these k vertices in the specific order? In this paper, we study k = 4. Let l be any even integer satisfying h(x₁, x₂) + h(x₂,x₃) + h(x₃, x₄) + h(x₄, x₁) ≤ I ≤ 2ⁿ. For n ≥ 5, we will prove that there exists a cycle C in Q_n of length I such that C traverses these 4 vertices in the specific order except for the case that I ∈ {6,8} when (x₁, x₃, x₂, x₄, x₁) forms a cycle of length 4.