Geodesic
Bivariate Chromatic Polynomials of Mixed Graphs
Discrete mathematics & theoretical computer science, Tome 25 (2023-2024) no. 2.

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The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.
DOI : 10.46298/dmtcs.9595
Classification : 05C31 
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     title = {Bivariate {Chromatic} {Polynomials} of {Mixed} {Graphs}},
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Beck, Matthias; Kolhatkar, Sampada. Bivariate Chromatic Polynomials of Mixed Graphs. Discrete mathematics & theoretical computer science, Tome 25 (2023-2024) no. 2. doi : 10.46298/dmtcs.9595. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.9595/

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