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    On the complexity of generalized chromatic polynomials

    Goodall, A. and Hermann, M. and Kotek, T. and Makowsky, J.A. and Noble, Steven (2018) On the complexity of generalized chromatic polynomials. Advances in Applied Mathematics 94 , pp. 71-102. ISSN 0196-8858.

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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, mcctmcct-colorings, and rainbow colorings, and many more. N. Linial (1986) showed that the chromatic polynomial χ(G;X)χ(G;X) is #P#P-hard to evaluate for all but three values X=0,1,2X=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 CP-colorings 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, mcctmcct-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: Article
    Keyword(s) / Subject(s): Graph polynomials, Counting complexity, Chromatic polynomial
    School: Birkbeck Schools and Departments > School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Administrator
    Date Deposited: 15 Jun 2017 09:04
    Last Modified: 26 Jul 2019 16:19
    URI: http://eprints.bbk.ac.uk/id/eprint/18934

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