A new two-variable generalization of the chromatic polynomial

Let P(G;x,y) be the number of vertex colorings φ : V →{1, ...,x} of an undirected graph G = (V,E) such that for all edges {u,v} ∈ E the relations φ(u)≤ y and φ(v)≤ y imply φ(u) 6= φ(v). We show that P(G;x,y) is a polynomial in x and y which is closely related to Stanley’s chromatic symmetric function, and which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of G. 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. Finally, we give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that P(G;x,y) can be evaluated in polynomial time for trees and graphs of restricted pathwidth.