Some corollaries of a theorem of Whitney on the chromatic polynomial

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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. © 1991.

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Lazebnik, F. (1991). Some corollaries of a theorem of Whitney on the chromatic polynomial. Discrete Mathematics, 87(1), 53–64. https://doi.org/10.1016/0012-365X(91)90070-I

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