Polychrome Labelings of Trees and Cycles

Abstract

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 $f^{\ast }(uv)=f(v)+f(w),uv\in E$, 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]}).