On certain classes of fractional matchings

Abstract An f-matching in an undirected graph X is defined as a set of vertex disjoint edges and odd cycles. In particular we consider f-matchings which saturate the maximum possible number of vertices and contain a maximum number of vertex disjoint edges. The main result is, that in this case different possible f-matchings in X with these properties contain the same number of triangles, pentagons and so on. This means that maximizing the set of vertex disjoint edges in the f-matchings determines the number of cycles of length 3 (i.e. triangles), 5,…,(2 n + 1). The problem is stated as a linear programming problem called fractional matching problem in a graph X .