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 (and probably of some practical value) 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. The concept of excessive [m]-factorization generalizes the concept of excessive factorization, which corresponds to the case in which m is the size of a perfect matching and was introduced in [A. Bonisoli and D. Cariolaro, Excessive factorizations of regular graphs, in Graph Theory in Paris, Trends in Math., Birkhäuser, Basel, 2007, 73–84]. MSC 2000: Primary 05C70; Secondary 05C15.