On the excessive [m]-index of a tree

The excessive [m]-index of a graph G is the minimum number of matchings of size m needed to cover the edge-set of G. We call a graph G [m]-coverable if its excessive [m]-index is finite. Obviously the excessive [1]-index is |E(G)| for all graphs and it is an easy task the computation of the excessive [2]-index for a [2]-coverable graph. The case m=3 is completely solved by Cariolaro and Fu in 2009. In this paper we prove a general formula to compute the excessive [4]-index of a tree and we conjecture a possible generalization for any value of m. Furthermore, we prove that such a formula does not work for the excessive [4]-index of an arbitrary graph.Comment: 12 pages, 7 figures, to appear in Discrete Applied Mathematic

Excessive [l,m]-factorizations

Given two positive integers l and m, with l 64m, an [l,m]-covering of a graph G is a set M of matchings of G whose union is the edge set of G and such that l 64;|M| 64m for every M. An [l,m]-covering M of G is an excessive [l,m]-factorization of G if the cardinality of M is as small as possible. The number of matchings in an excessive [l,m]-factorization of G (or 1e, if G does not admit an excessive [l,m]-factorization) is a graph parameter called the excessive [l,m]-index of G and denoted by \u3c7[l,m]\u2032(G). In this paper we study such parameter. Our main result is a general formula for the excessive [l,m]-index of a graph G in terms of other graph parameters. Furthermore, we give a polynomial time algorithm which computes \u3c7[l,m]\u2032(G) for any fixed constants l and m and outputs an excessive [l,m]-factorization of G, whenever the latter exists