  We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the
$H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic
symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically
equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study
uniqueness results for $H$-chromatic symmetric functions, with a particular
emphasis on the case $H$ is a complete bipartite graph. We also show that
several of the classical bases of the space of symmetric functions, i.e. the
monomial symmetric functions, power sum symmetric functions, and elementary
symmetric functions, can be realized as $H$-chromatic symmetric functions. We
end with some conjectures and open problems.
