On colored designs-III: on λ-colored H-designs, H having λ edges

Let H be a graph having λ edges which has no loops and multiple edges. Complete results about the existence of colored designs CH | CKnλ with exactly λ colors, are established for λ⩽4 and for arbitrary λ when H is a matching or a star. In all cases when there is a design it can be chosen to be cyclic. Slightly less complete results are obtained when H is a path or a cycle.