Klaus Dohmen ; André Poenitz ; Peter Tittmann - A new two-variable generalization of the chromatic polynomial

dmtcs:335 - Discrete Mathematics & Theoretical Computer Science, January 1, 2003, Vol. 6 no. 1 - https://doi.org/10.46298/dmtcs.335
A new two-variable generalization of the chromatic polynomial

Authors: Klaus Dohmen ; André Poenitz ; Peter Tittmann

    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.


    Volume: Vol. 6 no. 1
    Published on: January 1, 2003
    Imported on: March 26, 2015
    Keywords: chromatic polynomial,set partition,broken circuit,pathwidth,chromatic symmetric function,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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