Bounded color multiplicity graph isomorphism is in the #L hierarchy

In this paper we study the complexity of bounded color multiplicity graph isomorphism BCGI_b: the input is a pair of vertex-colored graphs such that the number of vertices of a given color in an input graph is bounded by b. We show that BCGI_b is in the #L hierarchy (more precisely, the Mod_kL hierarchy for some constant k depending on b). Combined with the fact that bounded color multiplicity graph isomorphism is logspace many-one hard for every set in the Mod_kL hierarchy for any constant k, we get a tight classification of the problem using logspace-bounded counting classes.