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
