Graphs and Combinatorics

, Volume 31, Issue 1, pp 91–98

On Weak Chromatic Polynomials of Mixed Graphs

  • Matthias Beck
  • Daniel Blado
  • Joseph Crawford
  • Taïna Jean-Louis
  • Michael Young
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 polynomial Mixed graph Poset ω-Labeling Order polynomial Combinatorial reciprocity theorem 

Mathematics Subject Classification (2000)

Primary 05C15 Secondary 05A15 06A07 

Copyright information

© Springer Japan 2013

Authors and Affiliations

  • Matthias Beck
    • 1
  • Daniel Blado
    • 2
  • Joseph Crawford
    • 3
  • Taïna Jean-Louis
    • 4
  • Michael Young
    • 5
  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA
  3. 3.Department of MathematicsMorehouse CollegeAtlantaUSA
  4. 4.Department of MathematicsAmherst CollegeAmherstUSA
  5. 5.Department of MathematicsIowa State UniversityAmesUSA

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