Let $n= 2k+1, |X|=|Y|= n$ and $G= (X, Y, E)$ be a $(k+1)$-regular bipartite graph.
Let `$M $`

be a perfect matching of `$G$`

having the property that every cycle of size 4
`$C_4 $`

intersects ` $M$ in at most one edge.

Does `$G $`

have a factorization ` $ M, M_1, M_2, ...., M_k$ such that:

for all $1\leq i \leq k$ ` $M \cup M_i$ is a Hamiltonian cycle