One Modulo N Gracefulness of Supersubdivision of Ladder

AbstractA 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 ɸ 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) ɸ is 1 − 1 (ii) ɸ induces a bijection ɸ* from the edge set of G to {1, N + 1, 2N + 1, …, N (q − 1) + 1} where ɸ*(uv) = ∣ɸ(u) − ɸ(v)∣. In this paper we prove that the Supersubdivision of Ladder is one modulo N graceful for all positive integers N.