  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.
