$n= 2k+1, |X|=|Y|= n$ 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
intersects $M$` in at most one edge.
$G $ have a factorization
$ M, M_1, M_2, ...., M_k$ such that:
$1\leq i \leq k$
$M \cup M_i$ is a Hamiltonian cycle