Magic valuations of finite graphs

The purpose of this paper is to investigate for graphs the existence of certain valuations which have some "magic" property. The question about the existence of such valuations arises from the investigation of another kind of valuations which are introduced in [1] and are related to cyclic decompositions of complete graphs into isomorphic subgraphs. Throughout this paper the word graph will mean a finite undirected graph without loops or multiple edges having at least one edge. By G ( m , n ) we denote a graph having m vertices and n edges, by V ( G ) and E ( G ) the vertex-set and the edge-set of G , respectively. Both vertices and edges are called the elements of the graph.