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On Weak Chromatic Polynomials of Mixed Graphs

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Abstract

A mixed graph is a graph with directed edges, called arcs, and undirected edges. A k-coloring of the vertices is proper if colors from {1, 2, . . . , k} are assigned to each vertex such that u and v have different colors if uv is an edge, and the color of u is less than or equal to (resp. strictly less than) the color of v if uv is an arc. The weak (resp. strong) chromatic polynomial of a mixed graph counts the number of proper k-colorings. Using order polynomials of partially ordered sets, we establish a reciprocity theorem for weak chromatic polynomials giving interpretations of evaluations at negative integers.

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Authors and Affiliations

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

    Matthias Beck

  2. Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, 91125, USA

    Daniel Blado

  3. Department of Mathematics, Morehouse College, Atlanta, GA, 30314, USA

    Joseph Crawford

  4. Department of Mathematics, Amherst College, Amherst, MA, 01002, USA

    Taïna Jean-Louis

  5. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Michael Young

Authors
  1. Matthias Beck
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  2. Daniel Blado
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  3. Joseph Crawford
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  4. Taïna Jean-Louis
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  5. Michael Young
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Correspondence to Matthias Beck.

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Beck, M., Blado, D., Crawford, J. et al. On Weak Chromatic Polynomials of Mixed Graphs. Graphs and Combinatorics 31, 91–98 (2015). https://doi.org/10.1007/s00373-013-1381-1

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  • DOI: https://doi.org/10.1007/s00373-013-1381-1

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