Integer k-matchings of graphs

An integer k-matching of a graph G is a function f that assigns to each edge an integer in {0,1,…,k} such that ∑e∈Γ(v)f(e)≤k for each v∈V(G). The k-matching number of G is the maximum number of ∑e∈E(G)f(e) over all k-matchings f. In this paper, when k is even, we give a relationship between some special fractional matchings and integer k-matchings, and then we obtain a formula for k-matching number by using fractional matching number and all the maximum integer k-matchings with the maximum number of edges assigned 0 (named 0-edges) can be constructed by using the algorithms given by Pulleyblank (1987) for generating some special fractional matchings. When k is odd, we obtain some properties of the maximum k-matchings with the maximum number of 0-edges.