Graceful Labeling of Balanced Stars of Paths

Abstract

Abstract If the vertices of a tree on n vertices are labeled by the distinct numbers l,2,…, n , in such a way that the set of edge-differences (i.e., the differences between the labels on adjacent pairs of vertices) is exactly {1, 2,…, n − 1}, then such a labeling is called a graceful labeling. The graceful tree conjecture of Ringel and Kotzig which states “every tree has a graceful labeling” still remains unresolved; only the graceful labeling of some particular classes of trees are known so far. [2] is an excellent survey on the state of knowledge in this field. The main result of this paper gives graceful labeling to a class of trees called balanced stars of paths , which are defined below.