Properties of Villarceau Torus

Abstract

Villarceau torus is a discrete graph theory model of spiral torus which is called Helical Toroidal Electron Model in Physics. It also represents the double stranded helix model of DNA. The spiral torus or toroidal helix in Physics and Molecular Biology is continuous whereas the Villarceau torus is discrete. The main contribution of the paper is the identification of cycles which are equivalent to toroidal helix in spiral torus. In addition, the poloidal revolution and the toroidal revolutions of Villarceau torus are computed. The paper identifies some striking differences between ring torus and Villarceau torus. It is proved that Villarceau torus does not admit any convex edgecuts and convex cycles other than 4-cycles. The convex cut method is extended to graphs which do not admit convex edgecuts. Using this technique, the network distance (Wiener Index) is computed, Also an optimal congestion-balanced routing for Villarceau torus is designed.