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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 $ always have a 1-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?

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Is there always a 1-factorization of such a graph, even without the extra requirement? –  Felix Goldberg Jul 23 at 14:14

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