Graphs and Combinatorics

, Volume 31, Issue 1, pp 91–98

On Weak Chromatic Polynomials of Mixed Graphs


    • Department of MathematicsSan Francisco State University
  • Daniel Blado
    • Department of Computing and Mathematical SciencesCalifornia Institute of Technology
  • Joseph Crawford
    • Department of MathematicsMorehouse College
  • Taïna Jean-Louis
    • Department of MathematicsAmherst College
  • Michael Young
    • Department of MathematicsIowa State University
Original Paper

DOI: 10.1007/s00373-013-1381-1

Cite this article as:
Beck, M., Blado, D., Crawford, J. et al. Graphs and Combinatorics (2015) 31: 91. doi:10.1007/s00373-013-1381-1


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.


Weak chromatic polynomialMixed graphPosetω-LabelingOrder polynomialCombinatorial reciprocity theorem

Mathematics Subject Classification (2000)

Primary 05C15Secondary 05A1506A07

Copyright information

© Springer Japan 2013