A generalized chromatic polynomial, acyclic orientations with prescribed sources and sinks, and network reliability

Abstract Suppose G=(V,E) is a graph and K, K′, K″ are subsets of V such that K⊆K′ ∩ K″. We introduce and study a polynomial P(G,K,K′,K″; λ) in λ. This polynomial coincides with the classical chromatic polynomial P(G; λ) when K=V. The results of this paper generalize Whitney's characterizations of the coefficients of P(G; λ) and the work of Stanley on acyclic orientations. Furthermore, we establish a connection between a family of polynomials associated with network reliability and a family of polynomials associated with P(G,K,K′,K″; λ).