Hamiltonian surfaces in the 4-cube, 4-bit Gray codes and Venn diagrams

Manuela Muzika Dizdarević, Rade T. Živaljević

We study Hamiltonian surfaces in the $d$-dimensional cube $I^d$ as intermediate objects useful for comparative analysis of Venn diagrams and Gray cycles. In particular we emphasize the importance of $0$-Hamiltonian spheres and the ``sphericity'' of Gray codes in the context of reducible Venn diagrams. For illustration we show that precisely two, out of the nine known types of $4$-bit Gray cycles, are not spherical. The unique, balanced Gray cycle is spherical, which in turn leads to a new construction of a reducible Venn diagram with 5 ellipses (originally constructed by P.~Hamburger and R.\,E.~Pippert).