ONE MODULO N GRACEFULNESS OF REGULAR BAMBOO TREE AND COCONUT TREE

A function f is called a graceful labelling of a graph G with q edges if f is an injection from the vertices of G to the set {0, 1, 2, . . . , q} such that, when each edge xy is assigned the label |f(x) − f(y)| , the resulting edge labels are distinct. A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function φ from the vertex set of G to {0, 1,N, (N + 1), 2N, (2N + 1), . . . ,N(q − 1),N(q − 1) + 1} in such a way that (i) φ is 1 − 1 (ii) φ induces a bijection φ_ from the edge set of G to {1,N + 1, 2N + 1, . . . ,N(q − 1) + 1} where φ_(uv)=|φ(u) − φ(v)| . In this paper we prove that the every regular bamboo tree and coconut tree are one modulo N graceful for all positive integers N .