Goodall, A. and Hermann, M. and Kotek, T. and Makowsky, J.A. and Noble, Steven (2017) On the complexity of Generalized Chromatic Polynomials. Technical Report. Birkbeck, University of London, London, UK.
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Abstract
J. Makowsky and B. Zilber (2004) showed that many variations of graph colorings, called CP-colorings in the sequel, give rise to graph polynomials. This is true in particular for harmonious colorings, convex colorings, mcct-colorings, and rainbow colorings, and many more. N. Linial (1986) showed that the chromatic polynomial �(G;X) is #P-hard to evaluate for all but three values X = 0, 1, 2, where evaluation is in P. This dichotomy includes evaluation at real or complex values, and has the further property that the set of points for which evaluation is in P is finite. We investigate how the complexity of evaluating univariate graph polynomials that arise from CPcolorings varies for different evaluation points. We show that for some CP-colorings (harmonious, convex) the complexity of evaluation follows a similar pattern to the chromatic polynomial. However, in other cases (proper edge colorings, mcct-colorings, H-free colorings) we could only obtain a dichotomy for evaluations at non-negative integer points. We also discuss some CP-colorings where we only have very partial results.
Metadata
| Item Type: | Monograph (Technical Report) |
|---|---|
| Additional Information: | Birkbeck Pure Mathematics Preprint Series #33 |
| Keyword(s) / Subject(s): | Graph polynomials, Counting Complexity, Chromatic Polynomial |
| School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |
| Depositing User: | Administrator |
| Date Deposited: | 22 Mar 2019 13:16 |
| Last Modified: | 09 Aug 2023 12:46 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/26760 |
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