Discrete Mathematics & Theoretical Computer Science, Vol 6, No 1 (2003)

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DMTCS vol 6 no 1 (2003), pp. 69-90

Discrete Mathematics & Theoretical Computer Science


Volume 6 n° 1 (2003), pp. 69-90

author:Klaus Dohmen and André Poenitz and Peter Tittmann
title:A new two-variable generalization of the chromatic polynomial
keywords:chromatic polynomial, set partition, broken circuit, pathwidth, chromatic symmetric function
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.

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reference: Klaus Dohmen and André Poenitz and Peter Tittmann (2003), A new two-variable generalization of the chromatic polynomial, Discrete Mathematics and Theoretical Computer Science 6, pp. 69-90
bibtex:For a corresponding BibTeX entry, please consider our BibTeX-file.
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Automatically produced on Fri Jun 20 22:18:04 CEST 2003 by falk