# 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