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={K1,...,KP} of vertex-disjoint odd circles and L a perfect matching in the graph obtained form G by removing K.Let ko:=|L| and 1∈N such that 21+1≤|V|<2(1+1)+1. By ki we denote the number of odd cycles of length 2i+1 containd in K (i=1,...,1). Then λ (F) :=(ko,k1,...,k1) 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.