  J. Przytycki has established a connection between the Hochschild homology of
an algebra $A$ and the chromatic graph homology of a polygon graph with
coefficients in $A$. In general the chromatic graph homology is not defined in
the case where the coefficient ring is a non-commutative algebra. In this paper
we define a new homology theory for directed graphs which takes coefficients in
an arbitrary $A-A$ bimodule, for $A$ possibly non-commutative, which on
polygons agrees with Hochschild homology through a range of dimensions.
