Abstract
Let \(\lambda \) be a positive integer. An acyclic \(\lambda \)-coloring of a digraph D is a partition of the vertices of D into \(\lambda \) color clases such that the color classes induce acyclic subdigraphs in D. The minimum integer \(\lambda \) for which there exists an acyclic \(\lambda \)-coloring of D is the dichromatic number dc(D) of D. Let \(P(D;\lambda )\) be the dichromatic polynomial of D, which is the number of acyclic \(\lambda \)-colorings of D. In this paper, a recursive formula for \(P(D;\lambda )\) is given. The coefficients of the polynomial \(P(D;\lambda )\) are studied. The dichromatic polynomial of a digraph D is related to the structure of its underlying graph UG(D). Also, we study dichromatic equivalently and dichromatically unique digraphs.
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Acknowledgements
We would like to thank the anonymous referees for their helpful remarks and suggestions to improve the presentation of the paper.
Funding
D. González-Moreno is supported by CONACyT-Mexico under projects CB-47510664 and CB-222104, and M Olsen is supported by CONACyT-Mexico under project CB-47510664.
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González-Moreno, D., Hernández-Ortiz, R., Llano, B. et al. The Dichromatic Polynomial of a Digraph. Graphs and Combinatorics 38, 85 (2022). https://doi.org/10.1007/s00373-022-02484-0
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DOI: https://doi.org/10.1007/s00373-022-02484-0