Simple, Reducible Venn Diagrams on Five Curves and Hamiltonian Cycles

Recently, using graph theory, we developed procedures for the construction of Venn diagrams. Utilizing these procedures with some new methods introduced here, we determine the number of simple, reducible spherical Venn diagrams of five sets. In so doing, we obtain examples of Venn diagrams which yield answers to several problems and conjectures of Grünbaum. Among others, we construct a simple, reducible Venn diagram with five congruent ellipses. We show that this diagram is unique on the sphere and produces two different plane diagrams. This corrects some erroneous statements that started with John Venn more than a century ago in 1880 and have been repeated frequently by others ever since.