On graceful trees

Abstract

A (p, q)-graph G = (V,E) is said to be (k, d)-graceful, where k and d are positive integers, if its p vertices admits an assignment of a labeling of numbers 0, 1, 2, ..., k + (q - 1)d such that the values on the edges defined as the absolute difference of the labels of their end vertices form the set {k, k + d, ..., k + (q - 1)d}. In this paper we prove that a class of trees called TP-trees and subdivision of TP- trees are (k, d)-graceful for all positive integers k and d.