Bivariate Chromatic Polynomials of Mixed Graphs [article]

Matthias Beck, Sampada Kolhatkar
2021 arXiv   pre-print
The bivariate chromatic polynomial χ_G(x,y) of a graph G = (V, E), introduced by Dohmen-Pönitz-Tittmann (2003), counts all x-colorings of G such that adjacent vertices get different colors if they are ≤ y. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main results is a decomposition formula which expresses χ_G(x,y) as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for χ_G(x,y).
arXiv:2111.09384v1 fatcat:qhoed2km7zdrned27dke53z6xq