Fat 4-polytopes ★★★
Author(s): Eppstein; Kuperberg; Ziegler
The fatness of a 4-polytope is defined to be where is the number of faces of of dimension .
Question Does there exist a fixed constant so that every convex 4-polytope has fatness at most ?
The Crossing Number of the Hypercube ★★
The crossing number of is the minimum number of crossings in all drawings of in the plane.
The -dimensional (hyper)cube is the graph whose vertices are all binary sequences of length , and two of the sequences are adjacent in if they differ in precisely one coordinate.
Conjecture
Keywords: crossing number; hypercube
The Crossing Number of the Complete Bipartite Graph ★★★
Author(s): Turan
The crossing number of is the minimum number of crossings in all drawings of in the plane.
Conjecture
Keywords: complete bipartite graph; crossing number