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arXiv · 1210.4634

On Weak Chromatic Polynomials of Mixed Graphs

Matthias Beck·Daniel Blado·Joseph Crawford·Taina Jean-Louis·Michael Young
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Abstract

A \emph{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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Matthias Beck, Daniel Blado, Joseph Crawford, Taina Jean-Louis, Michael Young. 2012-10-17. On Weak Chromatic Polynomials of Mixed Graphs. https://arxiv.org/abs/1210.4634

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