Polychrome Labelings of Trees and Cycles

This paper deals with a new kind of graph labeling similar to the well known harmonious, graceful, and elegant labelings. A polychrome labeling of a simple and connected graph \(G = (V, E)\) by an abelian group \(A\) is a bijective map from \(V\) onto \(A\) such that the induced edge labeling \(f^*(uv) = f(v) + f(w), uv \in E\), is injective. Polychrome labelings of a path and a cycle by a large class of abelian groups are designed, and the connection to the above mentioned labelings is shown. In addition, the author presents a conjecture which is similar to a famous conjecture of G. Ringel about graceful trees (see {[9]}).