Covering graphs with matchings of fixed size

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.