The Birkhoff-Lewis Equations for Graph-Colorings

The paper Chromatic Polynomials by G.D. Birkhoff and D.C. Lewis has long served as a textbook for students of map colorings. Much of it is concerned with equations relating to two kinds of chromatic polynomial (or chromial), the free and the constrained. In a paper in Discrete Mathematics the author simplified the theory of these equations and obtained what can charitably be described as a general solution. (It requires the inversion of a very large matrix, denoted in this paper by M). He reported on this work at the conference Quo Vadis, Graph Theory? The present paper is based on his lecture notes. Essentially it is a shortened version of the paper in Discrete Mathematics but there are differences of approach. This paper keeps closer to the spirit of the grand original by relating constrained chromials directly to free, whereas the paper in Discrete Mathematics proceeds by relating non-planar free chromials to planar ones.