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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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
Keywords
- Weak chromatic polynomial
- Mixed graph
- Poset
- ω-Labeling
- Order polynomial
- Combinatorial reciprocity theorem