Embedding Cycles into Hypercubes with Prescribe Vertices in the Specific Order

In this paper, we are interesting in a new cycle embedding problem. Let ${\bf x_1}, {\bf x_2}, \ldots, {\bf x_k}$ 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 $h({\bf x_1}, {\bf x_2}) + h({\bf x_2}, {\bf x_3}) + h({\bf x_3}, {\bf x_4}) + h({\bf x_4}, {\bf x_1}) \le l \le 2^n$. For $n \geq 5$, we will prove that there exists a cycle $C$ in $Q_n$ of length $l$ such that $C$ traverses these $4$ vertices in the specific order except for the case that $l\in \{6,8\}$ when $\langle {\bf x_1}, {\bf x_3}, {\bf x_2}, {\bf x_4}, {\bf x_1}\rangle$ forms a cycle of length 4.