I have no idea how to prove this one. It looks like common sense, but I don't know hot to proceed with the proof.
2026-03-28 11:35:19.1774697719
Prove that, if all perfect matchings in G are pairwise disjoint, than every two perfect matchings contain the edge set of a hamiltonian cycle in G.
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Think it through in two steps:
Because if you can, then $M_3$ is certainly not disjoint from either $M_1$ or $M_2$, and so not all matchings are pairwise disjoint.
Conversely, if all matchings are pairwise disjoint, then we can never find this $M_3$, which means that $M_1 \cup M_2$ must be a Hamiltonian cycle.