Enumerating the total colorings of a polyhedron and application to polyhedral links

Abstract

In this paper we consider the enumeration problem of a particular three-dimensional molecular or chemical compound system which has a polyhedral frame where the vertices, edges and faces represent 'units' such as atoms, bonds, ligands, polymers, or other objects of chemical interests. In this system, chirality is also taken into account. This enumeration problem is mathematically modeled as the 'total coloring' enumeration problem of a polyhedron: i.e., the number of ways to color all the vertices, edges and faces of the polyhedron by using three or more corresponding color sets, in which some colors may be chiral. We establish a general formula for this enumeration problem by extending the fundamental version of Plya's enumeration theorem. In particular, we apply this technique to the enumeration problem of polyhedral links which have received special attention from biochemists, mathematical chemists and mathematicians over the past two decades.NSFC [10831001