Some corollaries of a theorem of Whitney on the chromatic polynomial

Abstract

Let F denote the family of simple undirected graphs on v vertices having e edges, P( G; λ) be the chromatic polynomial of a graph G. For the given integers v, e, λ, let f( v, e, λ)= max\\s{ P( G;λ): G∈ F \\s}. In this paper we determine some lower and upper bounds for f( v, e, λ) provided that λ is sufficiently large. In some cases f( v, e, λ) is found and all graphs G for which P( G; λ) = f( v, e, λ) 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; λ) in terms of the number of ‘broken circuits’ in G.