SOME COROLLARIES OF A THEOREM OF WHITNEY ON THE CHROMATIC POLYNOMIAL

Let F denote the family of simple undirected graphs on upsilon vertices having e edges, P(G; lambda) be the chromatic polynomial of a graph G. For the given integers upsilon, e, lambda, let closed-integral(upsilon, e, lambda) = max {P(G; lambda): G is-a-member-of F. In this paper we determine some lower and upper bounds for closed-integral(upsilon, e, lambda) provided that lambda is sufficiently large. In some cases closed-integral(upsilon, e, lambda) is found and all graphs G for which P(G; lambda) = closed-integral(upsilon, e, lambda) are described. Connections between these problems and some other questions from the extremal graph theory are analysed using Whitney's characterization of the coefficients of P(G; lambda) in terms of the number of 'broken circuits' in G.