On the conjecture of bijection between perfect matching and sub-hypercube in folded hypercubes

Dong and Wang in [Theor. Comput. Sci. 771 (2019) 93–98] conjectured that the resulting graph of the $n$-dimensional folded hypercube $F{Q}_n$ by deleting any perfect matching is isomorphic to the hypercube $Q_n$. In this paper, we show that the conjecture holds when $n=2,3$, and it is not true for $n\ge 4$.