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A Bivariate Chromatic Polynomial for Signed Graphs

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Abstract

We study Dohmen–Pönitz–Tittmann’s bivariate chromatic polynomial \(c_\Gamma (k,l)\) which counts all \((k+l)\)-colorings of a graph \(\Gamma \) such that adjacent vertices get different colors if they are \(\le k\). Our first contribution is an extension of \(c_\Gamma (k,l)\) to signed graphs, for which we obtain an inclusion–exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for \(c_\Gamma (k,l)\) and its signed-graph analogues, reminiscent of Stanley’s reciprocity theorem linking chromatic polynomials to acyclic orientations.

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Authors and Affiliations

  1. Department of Mathematics, San Francisco State University, 1600 Holloway Ave, San Francisco, CA, 94132, USA

    Matthias Beck

  2. School of Mathematical and Statistical Sciences, Arizona State University, Physical Sciences Building A-Wing Rm. 216, 901 S. Palm Walk, Tempe, AZ, 85287, USA

    Mela Hardin

Authors
  1. Matthias Beck
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  2. Mela Hardin
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Correspondence to Matthias Beck.

Additional information

We thank the referees for helpful suggestions. This research was partially supported by the U. S. National Science Foundation through the grants DMS-1162638 (Beck) and DGE-0841164 (Hardin).

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Beck, M., Hardin, M. A Bivariate Chromatic Polynomial for Signed Graphs. Graphs and Combinatorics 31, 1211–1221 (2015). https://doi.org/10.1007/s00373-014-1481-6

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  • DOI: https://doi.org/10.1007/s00373-014-1481-6

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