One modulo N gracefulness of splitting graphs and subdivision of double triangle graphs
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 S’(P2n) , S’(P2n+1) , S’(K1,n) , all subdivision of double triangular snakes (2Δk-snake) and all subdivision of 2mΔk-snake are one modulo N graceful for all positive integers N.