Excessive[l,m]-factorizations

Abstract

MatchingChromatic index a b s t r a c t Given two positive integers l and m, with l ≤ m, 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 ≤ |M| ≤ m for everythe cardinality of M is as small as possible. The number of matchings in an excessive [l, m]-factorization of G (or ∞, 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 χ ′ [l,m] (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 χ ′ [l,m] (G) for any fixed constants l and m and outputs an excessive [l, m]-factorization of G, whenever the latter exists.