Abstract:

Two new labelings related to the odd-graceful problem are introduced. A necessary and sufficient condition between a bipartite odd-graceful tree and a bipartite graceful tree is shown. Several constructive methods for constructing large scale of odd-graceful trees are determined. It has been shown that a spider, each of leg which has length in ｛m,m+1｝ for m≥1, is odd-graceful.

Key words: odd-graceful labeling; strongly odd-graceful trees; (strongly) bipartite odd-graceful trees