Expansions of the chromatic polynomial

The chromatic polynomial (or chromial) of a graph was first defined by Birkhoff in 1912, and gives the number of ways of properly colouring the vertices of the graph with any number of colours. A good survey of the basic facts about these polynomials may be found in the article by Read 3].It has recently been noticed that some classical problems of physics can be expressed in terms of chromials, and papers by Nagle 2], Baker 1], Temperley and Lieb 4], are concerned with methods of expanding the chromial for use in such problems. In this note we shall unify, simplify, and generalise their treatments, confining our attention to the theoretical basis of the methods.