Zyklische Orientierungen endlicher bewerteter Graphen

Abstract

Starting from problem 4 ofK. Wagner [2],H. Fleischner andP. D. Vestergaard [1] introduce the notion of a value-true walk in a finite, connected graph, the edges of which are valuated with nonnegative integers. Their main theorem states that the existence of such a walk is equivalent to the existence of an orientation of the edges with the following property: For every vertex the sum of the valuations of the incoming edges equals the sum of the valuations of the outgoing edges. Let us call such an orientation a cyclic one. In the present paper we study finite, valuated graphs that admit a cyclic orientation. First, we give two necessary conditions for a valuated graphG to admit a cyclic orientation concerning the stars and the bonds ofG, respectively. (The starS (v) of a vertexv is the set of all edges ofG incident withv.) Then, as the main part of the paper we give a characterization of those graphs for which the star- and the bond-condition is sufficient, respectively (for any valuation of the graph). These characterizations are in terms of constructability from trees andK ₃, respectively, as well as in terms of forbidden subgraphs.