Algorithmic bounds on the chromatic number of a graph

Abstract

The chromatic number of a graph is the smallest number of colors required to color its vertices such that no two adjacent vertices share a color. In the general case a problem of determining the chromatic number is NP-hard, thus any graph invariants that can be used to bound it are of great interest. Within this paper we discuss the properties of the invariants originating in the notion of a potential function. We study their interdependencies and the relationships to the classical Welsh-Powell and Szekeres-Wilf numbers. We also present the results of experimental comparison of two known sequential algorithms to the algorithms that use orderings of vertices with respect to their potentials.