Cubical 4-polytopes with few vertices (Q1360959)

It is well-known that a convex polytope \(P\), every facet of which is a combinatorial cube, is said to be a cubical polytope. On the other hand, a \(d\)-dimensional polytope is called almost simple if each vertex of its graph is either \(d\)-valent or \((d+1)\)-valent. A known result is the following: for \(d>4\) all but one cubical \(d\)-polytopes with up to \(2^{d+1}\) vertices are almost simple, providing a complete enumeration of all the cubical \(d\)-polytopes with up to \(2^{d+1}\) vertices. Using shelling sequences of the boundary of \(P\) in combination with graph-theoretic arguments, the authors prove that the above mentioned result is also true if \(d=4\).