A Contribution to the Theory of Chromatic Polynomials

Two polynomials 6(G, n) and <f>(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 <f>(G, n). Attention is called to some unsolved problems concerning <t>(G, n) which are natural generalizations of the Four Colour Problem from planar graphs to general graphs. A polynomial x(G, x, y) in two variables x and y, which can be regarded as generalizing both 0(G, n) and <f>(G, n) is studied. For a connected graph x(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. The invariance of x(G, x, y) under a change of this enumeration is apparently a new result about spanning trees. I t is observed that the theory of spanning trees now links the theory of graphcolourings to that of electrical networks.