One Modulo N Gracefullness Of Arbitrary Supersubdivisions of Graphs

Abstract: 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 arbitrary supersubdivisions of paths, disconnected paths, cycles and stars are one modulo N graceful for all positive integers N