# F-factors, perfect matchings and related concepts

## Abstract

The concept of an F-factor of a graph G=(V,E) has been introduced by MÜHLBACHER. Here an F-factor F=L U K is a collection K={K_{1},...,K_{P}} of vertex-disjoint odd circles and L a perfect matching in the graph obtained form G by removing K.

Let k_{o}:=|L| and 1∈N such that 21+1≤|V|<2(1+1)+1. By k_{i} we denote the number of odd cycles of length 2i+1 containd in K (i=1,...,1). Then λ (F) :=(k_{o},k_{1},...,k_{1}) is called the characteristic vector of F.

In this paper we will present some results on the relationship between perfect matchings and F-factors. We will show that the problem of finding an F-factor in G can be transformed into an equivalent maximum cardinality matching problem. Since MÜHLBACHER showed how to construct a maximum cardinality matching from an F-factor, both problems are of the same complexity.

We will show some criteria for the existence of F-factors and deduce an algorithm for solving the problem of determining a canonical F-factor.

Here an F-factor F is called canonical if λ (F)≥λ (F′) for all F-factors F′.

To our knowledge such an algorithm was not known before.

## Preview

Unable to display preview. Download preview PDF.

## 6. References

- [1]Balinski,M.: Integer Programming: Methods, Uses, Computation, in: G.B.Dantzig and A.F.Veinott (eds.): Mathematics of the Decision Sciences. Part I. 179–256. Providence: American Mathematical Society (1968).Google Scholar
- [2]Berge, C.: Two Theorems in Graph Theory. Proc. Natl. Acad. Sci. U. S., 43, (1957), 842–844.Google Scholar
- [3]Burkard, R.E., W. Hahn and U. Zimmermann: An Algebraic Approach to Assignment Problems. Math. Progr. 12, (1977), 318–327.Google Scholar
- [4]Edmonds, J.: Path, Trees, and Flowers. Can. J. Math. 17, (1965a), 449–467.Google Scholar
- [5]Edmonds,J.: Maximum Matching and a Polyhedorn with 0,1 Vertices. J. Res. NBS, (1965b), 125–130.Google Scholar
- [6]Hopcroft, J. und R.M. Karp: An N
^{5/2}Algorithm for Maximum Matchings in Bipartite Graphs. SIAM Journal on Computing 2, (1973), 225–231Google Scholar - [7]Mühlbacher, J.: F-Factors of Graphs: A Generalized Matching Problem, Information Processing Letters, 8, (1979), 207–214.Google Scholar
- [8]Nemhauser, G.L. and L.E. Trotter, Jr.: Properties of Vertex Packing and Independence System Polyhedra. Math. Progr. 6, (1974), 48–61.Google Scholar