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

## TL;DR

This paper investigates a conjecture relating perfect matchings and sub-hypercubes in folded hypercubes, confirming it for small dimensions but disproving it for larger ones, thus clarifying the structure of these graphs.

## Contribution

The paper proves the conjecture holds for dimensions 2 and 3, and demonstrates it does not hold for dimensions 4 and above, providing new insights into folded hypercube structures.

## Key findings

- Conjecture holds for n=2,3.
- Conjecture does not hold for n≥4.
- Clarifies the structure of folded hypercubes after perfect matching removal.

## Abstract

Dong and Wang in [Theor. Comput. Sci. 771 (2019) 93--98] conjectured that the resulting graph of the $n$-dimensional folded hypercube $FQ_n$ 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 $n\geq4$.