1 Introduction
A polycube is an object constructed by gluing cubes wholeface to wholeface, such that its surface is a manifold. Thus the neighborhood of every surface point is a disk; so there are no edgeedge nor vertexvertex nonmanifold surface touchings. Here we only consider polycubes of genus zero. The edges of a polycube are all the cube edges on the surface, even when those edges are shared between two coplanar faces. Similarly, the vertices of a polycube are all the cube vertices on the surface, even when those vertices are flat, incident to face angles. Such polycube flat vertices are degree. It will be useful to distinguish these flat vertices from corner vertices, nonflat vertices with incident angles (degree, , or ). For a polycube , let its skeleton graph include every vertex and edge of , with vertices marked as either corner or flat.
It is an open problem to determine whether every polycube has an edgeunfolding, a tree in the skeleton that spans all corner vertices (but need not include flat vertices), which, when cut, unfolds the surface to a net, a planar, nonoverlapping polygon [O’R19]. Here by nonoverlapping is meant that no two points, each interior to a face, are mapped to same point in the plane. This allows two boundary edges to coincide in the net; so the polygon is “weakly simple.” The intent is that we want to be able to cut out the net and refold to . Henceforth “edgeunfolding” will mean: an edgeunfolding to a net.
It would be remarkable if it were true that every polycube could be edgeunfolded, but no counterexample is known. There has been considerable exploration of orthogonal polyhedra, a more general type of object, for which there are examples that cannot be edgeunfolded [BDD98]. (See [DF18] for citations to earlier work.) But polycubes have more edges in their skeleton graphs for the cut tree to follow than do orthogonal polyhedra, so it is conceivably easier to edgeunfold polycubes.
A restriction of edgeunfolding has been studied in [DDL10], [O’R10], [DDU13]: edgeunzipping. This is an edgeunfolding whose cut tree is a path (so that the surface could be “unzipped”). It is apparently unknown if even this highly restricted edgeunzipping could unfold every polycube to a net. The result of this note is to settle this question in the negative: two different polycubes are constructed each of which has no edgeunzipping. They are shown in Figure 1 and will be described later.
2 Hamiltonian Paths
Shephard [She75] introduced Hamiltonian unfoldings of convex polyhedra, what we are now calling edgeunzippings, following the terminology of [DDL10].^{1}^{1}1 “Unzipping” is a slight variation on their “zipper unfoldings.” It is easy to see that not every convex polyhedron has an edgeunzipping, simply because the rhombic dodecahedron has no Hamiltonian path. This counterexample avoids confronting the difficult nonoverlapping condition. We follow a similar strategy here, constructing a polycube with no Hamiltonian path. But there is a difference in that a polycube edgeunzipping need not include flat vertices, and so need not be a Hamiltonian path in . Thus identifying a polycube that has no Hamiltonian path does not immediately establish that has no edgeunzipping, if has flat vertices.
So one approach is to construct a polycube that has no flat vertices—every vertex is a corner vertex. Then if has no Hamiltonian path, then it has no edgeunzipping. A natural candidate is the polycube object shown in Fig. 2.
However, the skeleton of does admit Hamiltonian paths, and indeed we found a path that unfolds to a net.
Let be the dual graph of : each cube is a node, and two nodes are connected if they are glued facetoface. A polycube tree is a polycube whose dual graph is a tree. is a polycube tree. That it has a Hamiltonian path is an instance of a more general claim:
Lemma 1
The graph for any polycube tree has a Hamiltonian cycle.
Proof: It is easy to see by induction that every polycube tree can be built by gluing cubes each of which touches just one face at the time of gluing: never is there a need to glue a cube to more than one face of the previously built object.
A single cube has a Hamiltonian cycle. Now assume that every polycube tree of cubes has a Hamiltonian cycle. For a tree of cubes, remove a leafnode cube , and apply the induction hypothesis. The exposed square face to which glues to make includes either or edges of the Hamiltonian cycle ( would close the cycle; or would imply the cycle misses some vertices of ). It is then easy to extend the Hamiltonian cycle to include , as shown in Figure 3.
So to prove that a polycube tree has no edgeunzipping would require an argument that confronted nonoverlap. This leads to an open question:
Question 1
Does every polycube tree have an edgeunzipping?
3 Bipartite
To guarantee the nonexistence of Hamiltonian paths, we can exploit the bipartiteness of , using Lemma 2 below.
Lemma 2
A polycube graph is colorable, and therefore bipartite.
Proof: Label each lattice point of with the parity of the sum of the Cartesian coordinates of . A polycube ’s vertices are all lattice points of . This provides a coloring of ; colorable graphs are bipartite.
The parity imbalance in a colored (bipartite) graph is the absolute value of the difference in the number of nodes of each color.
Lemma 3
A bipartite graph with a parity imbalance has no Hamiltonian path.^{2}^{2}2 Stated at http://mathworld.wolfram.com/HamiltonianPath.html.
Proof: The nodes in a Hamiltonian path alternate colors . Because by definition a Hamiltonian path includes every node, the parity imbalance in a bipartite graph with a Hamiltonian path is either (if of even length) or
(if of odd length).
So if we can construct a polycube that (a) has no flat vertices, and (b) has parity imbalance , then we will have established that has no Hamiltonian path, and therefore no edgeunzipping. We now show that the polycube , illustrated in Figure 4, meets these conditions
Lemma 4
The polycube ’s graph has parity imbalance of .
Proof: Consider first the cube that is the core of ; call it . The front face has an extra ; see Fig. 5.
It is clear that the corners of are all colored . The midpoint vertices of the edges of are colored . Finally the face midpoints are colored . So vertices are colored and colored .
Next observe that attaching a cube to exactly one face of any polycube does not change the parity: the receiving face has colors , and the opposite face of has colors .
Now, can be constructed by attaching six copies of a cube “cross,” call it , which in isolation is a polycube tree and so can be built by attaching cubes each to exactly one face. And each attaches to one corner cube of . Therefore retains ’s imbalance of .
The point of the attachments is to remove the flat vertices of . Note that when attached to , each has only corner vertices.
Theorem 1
There is no edgeunzipping of .
4 Construction of
It turns out that the smaller polycube shown in Figure 6 also has no edgeunzipping, even though it has flat vertices.
To establish this, we still need an imbalance , which easily follows just as in Lemma 4:
Lemma 5
The polycube ’s graph has parity imbalance of .
But notice that has three flat vertices: , , and .
Theorem 2
There is no edgeunzipping of .
Proof: An edgeunzipping need not pass through the three flat vertices, , , and , but it could pass through one, two, or all three. We show that in all cases, an appropriately modified subgraph of has no Hamiltonian path. Let be a hypothetical edgeunzipping cut path. We consider four exhaustive possibilities, and show that each leads to a contradiction.
 (0) includes .
 (1) excludes one flat vertex and includes .

(Because of the symmetry of , it is no loss of generality to assume that it is that is excluded.) If excludes , then it does not travel over any of the four edges incident to . Thus we can delete from ; say that . This graph is shown in Fig. 7.
 (2) includes just one flat vertex , and excludes .

(Again symmetry ensures there is no loss of generality in assuming the one included flat vertex is .) must include corner , which is only accessible in through the three flat vertices. If excludes , then it must include the edge . Let . In , has degree , so terminates there. It must be that is a Hamiltonian path in , but the deletion of increases the parity imbalance to , and so again such a Hamiltonian path cannot exist.
 (3) excludes .

Because corner is only accessible through one of these flat vertices, never reaches and so cannot be an edgeunzipping
Thus the assumption that there is an edgeunzipping path for reaches a contradiction in all four cases. Therefore, there is no edgeunzipping path for .^{3}^{3}3 Just to verify this conclusion, we constructed these graphs in Mathematica and FindHamiltonianPath[] returned {} for each.
5 Edgeunfoldings of and
Now that it is known that and each have no edgeunzipping, it is natural to wonder if either settles the edgeunfolding open problem: Can they be edgeunfolded? Indeed both can: see Figures 8
Figure 10 shows a partial folding of , and animations are at http://cs.smith.edu/~jorourke/Unf/NoEdgeUnzip.html.
Acknowledgements.
We thank participants in the Bellairs 2018 workshop for their insights.
References
 [BDD98] Therese Biedl, Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Joseph O’Rourke, Mark Overmars, Steve Robbins, and Sue Whitesides. Unfolding some classes of orthogonal polyhedra. In Proc. 10th Canad. Conf. Comput. Geom., pages 70–71, 1998. Full version in Elec. Proc.: http://cgm.cs.mcgill.ca/cccg98/proceedings/cccg98biedlunfolding.ps.gz.
 [DDL10] Erik D. Demaine, Martin L. 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.
 [DDU13] Erik D. Demaine, Martin L. Demaine, and Ryuhei Uehara. Zipper unfoldability of domes and prismoids. In Proc. 25th Canad. Conf. Comput. Geom., August 2013.
 [DF18] Mirela Damian and Robin Flatland. Unfolding orthotrees with constant refinement. http://arxiv.org/abs/1811.01842, 2018.
 [GDG18] Amanda Ghassaei, Erik D. Demaine, and Neil Gershenfeld. Fast, interactive origami simulation using GPU computation. In Origami: 7th Internat. Mtg. Origami Science, Mathematics and Education (OSME), 2018.
 [O’R10] Joseph O’Rourke. Flat zipperunfolding pairs for Platonic solids. http://arxiv.org/abs/1010.2450, October 2010.
 [O’R19] Joseph O’Rourke. Unfolding polyhedra. In Proc. 31st Canad. Conf. Comput. Geom., August 2019.
 [She75] Geoffrey C. Shephard. Convex polytopes with convex nets. Math. Proc. Camb. Phil. Soc., 78:389–403, 1975.