I want to pack n cubes in 3-space under the following 3 restrictions:
1) At each vertex only 2 cubes may touch
2) No two cubes may share an edge
3) No two cubes share any subface
2,3 just mean that they have to touch on a vertex
I am interested in minimizing the number of corners of the cubes which don't touch any other cubes.
Does anyone have an idea for a lowerbound on the number of vertices which only belong to one cube?
In other words, regardless of the arrangement, how many vertices will only belong to one cube?
The only lowerbound I have is the trivial 8.