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Does every convex polyhedron P have a zipper unfolding? A zipper unfolding cuts open P via a single path, necessarily a Hamiltonian path (to span all vertices), and unfolds the surface to a non-overlapping polygon in the plane. The segments of the path need not lie along edges of P.
Posed as Open Problem 2 in [DDL+10], which introduced the term “zipper unfolding.”
With the restriction that the cuts follow edges, any P without a Hamiltonian path in its 1-skeleton has no zipper edge-unfolding, e.g., a rhombic dodecahedron. (Such polyhedra have been studied, e.g., in [Bro61].)
J. O’Rourke, 7 Feb. 2012.
Thomas Brown. Simple paths on convex polyhedra. Pacific J. Math., 11(4):1211–1241, 1961.
Erik Demaine, Martin Demaine, Anna Lubiw, Arlo Shallit, and Jonah Shallit. Zipper unfoldings of polyhedral complexes. In Proc. 22nd Canad. Conf. Comput. Geom., pages 219–222, August 2010.