Abstract
We 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.
Highlights
All graphs in this paper are assumed to be finite, undirected and without loops or multiple edges
We propose a different generalization of the chromatic polynomial by weakening the requirements for proper colorings: Let X = Y ∪ Z, Y ∩ Z = 0/, be the set of available colors with |X | = x and |Y | = y
We give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that our generalized chromatic polynomial can be evaluated in polynomial time for trees and graphs of restricted pathwidth
Summary
All graphs in this paper are assumed to be finite, undirected and without loops or multiple edges. The well-known chromatic polynomial P(G; y) of a graph G = (V, E) gives the number of vertexcolorings of G with at most y colors such that adjacent vertices receive different colors. We propose a different generalization of the chromatic polynomial by weakening the requirements for proper colorings: Let X = Y ∪ Z, Y ∩ Z = 0/ , be the set of available colors with |X | = x and |Y | = y. The number of generalized proper colorings of G is denoted by. P(G; x, y), and as we will see in Theorem 1 below, this number turns out to be a polynomial in x and y, which we refer to as the generalized chromatic polynomial of G. We give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that our generalized chromatic polynomial can be evaluated in polynomial time for trees and graphs of restricted pathwidth
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