Covering graphs with matchings of fixed size

Let mm be a positive integer and let GG be a graph. We consider the question: can the edge set E(G)E(G) of GG be expressed as the union of a set MM of matchings of GG each of which has size exactly mm? If this happens, we say that GG is [m][m]-coverable   and we call MM an [m][m]-covering   of GG. It is interesting to consider minimum  [m][m]-coverings, i.e. [m][m]-coverings containing as few matchings as possible. Such [m][m]-coverings will be called excessive  [m][m]-factorizations  . The number of matchings in an excessive [m][m]-factorization is a graph parameter which will be called the excessive  [m][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.