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Problem 77: Zipper Unfoldings of Convex Polyhedra
 Statement
 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 nonoverlapping polygon in the plane.
The segments of the path need not lie along edges of P.
 Origin
 Posed as Open Problem 2 in [DDL+10], which
introduced the term "zipper unfolding."
 Status/Conjectures
 Open.
 Partial and Related Results
 With the restriction that the cuts follow edges,
any P without a Hamiltonian path in its 1skeleton
has no zipper edgeunfolding, e.g., a rhombic dodecahedron.
(Such polyhedra have been studied, e.g., in [Bro61].)
 Related Open Problems
 Problem 9.
 Categories
 polyhedra
 Entry Revision History
 J. O'Rourke, 7 Feb. 2012.
 Bro61

Thomas Brown.
Simple paths on convex polyhedra.
Pacific J. Math., 11(4):12111241, 1961.
 DDL+10

Erik Demaine, Martin Demaine, Anna Lubiw, Arlo Shallit, and Jonah Shallit.
Zipper unfoldings of polyhedral complexes.
In Proc. 22nd Canad. Conf. Comput. Geom., pages 219222,
August 2010.
The Open Problems Project  September 19, 2017