Characterization of Hamilton-path extendability for matchings

Characterize precisely when a matching M of Q_n can be extended to a Hamilton path between prescribed opposite-parity vertices u and v, by proving that such an extension exists if and only if none of the three C-conditions holds: M contains a half-layer covering both endpoints; M contains a u-avoiding almost half-layer in the direction from u to v together with the specified matching edges incident with the neighboring vertices; or uv belongs to M.

Background

The paper identifies three structural obstructions, called C1–C3, and proves that each obstruction is necessary: whenever one holds, no Hamilton path with the prescribed endpoints can extend the matching. The unresolved part is sufficiency for arbitrary matchings in Q_n with n ≥ 5. The result is proved for matchings spanning at most five directions.