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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 non-overlapping 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 1-skeleton has no zipper edge-unfolding, 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.

Bibliography

Bro61
Thomas Brown.
Simple paths on convex polyhedra.
Pacific J. Math., 11(4):1211-1241, 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 219-222, August 2010.



The Open Problems Project - December 04, 2015