Earlier Helme Guizon and Rong have defined chromatic graph homology complex for a graded commutative algebra, and it is easy to extend the definition to graded commutative DG algebra. One of important applications, considered earlier in our joint work with Radmila Sazdanovic, is to the case of an algebra computing cohomology of a manifold, such as the de Rham Complex. However, when working over Z or in finite field, it is desirable to work with algebras which are commutative up to homotopy. I am planning to explain our recent work with Maksym Zubkov, where we construct such a homology complex, and also to outline further questions related to it.