https://hal.inria.fr/hal-00958990Dohmen, KlausKlausDohmenFaculty Mathematics / Sciences / Computer Science [Mittweida] - Hochschule Mittweida - University of Applied SciencesPoenitz, AndréAndréPoenitzFaculty Mathematics / Sciences / Computer Science [Mittweida] - Hochschule Mittweida - University of Applied SciencesTittmann, PeterPeterTittmannFaculty Mathematics / Sciences / Computer Science [Mittweida] - Hochschule Mittweida - University of Applied SciencesA new two-variable generalization of the chromatic polynomialHAL CCSD2003chromatic polynomialset partitionbroken circuitpathwidthchromatic symmetric function[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Inria Sophia Antipolis-Méditerranée / I3s, Service Ist2014-03-13 16:59:102019-05-07 11:26:012014-03-13 21:14:02enJournal articleshttps://hal.inria.fr/hal-00958990/document10.46298/dmtcs.335application/pdf1We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.