A Contribution to the Theory of Chromatic Polynomials

Abstract

Two polynomials θ ( G, n ) and ϕ ( G, n ) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial ϕ ( G, n ). Attention is called to some unsolved problems concerning ϕ ( G, n ) which are natural generalizations of the Four Colour Problem from planar graphs to general graphs. A polynomial χ ( G, x, y ) in two variables x and y , which can be regarded as generalizing both θ ( G, n ) and ϕ ( G, n ) is studied. For a connected graph χ ( G, x, y ) is defined in terms of the “spanning” trees of G (which include every vertex) and in terms of a fixed enumeration of the edges.