Bivariate Chromatic Polynomials of Mixed Graphs
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 an...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2023-11-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | http://dmtcs.episciences.org/9595/pdf |
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| Summary: | 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 $\chi_G(x,y)$. |
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| ISSN: | 1365-8050 |