Let m be a positive integer and let G be a graph. We consider the question: can the edge set E (G) of G be expressed as the union of a set M of matchings of G each of which has size exactly m? If this happens, we say that G is [m]-coverable and we call M an [m]-covering of G. It is interesting to consider minimum[m] -coverings, i.e. [m]-coverings containing as few matchings as possible. Such [m]-coverings will be called excessive[m] -factorizations. The number of matchings in an excessive [m]-factorization is a graph parameter which will be called the excessive[m] -index and denoted by χ[m]′. In this paper we begin the study of this new parameter as well as of a number of other related graph parameters.