Colligation of cycle graphs on one modulo N graceful labelling and its applications

Abstract

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) = |f (u) – f (v)|. In this paper to propose that the generalization of cycle related graphs on one modulo N graceful labelling.The following are the findings regarding cycles(i) Cn for n ≡ 3 (mod 4) is graceful but not one modulo N graceful, for N > 1.(ii) Cn for n ≡ 2 (mod 4) is odd graceful but neither graceful nor one modulo N graceful, for N ≥ 3.(iii) Cn for n ≡ 1 (mod 4) is not one modulo N graceful, for any positive integer N.(iv) Cn for n ≡ 0 (mod 4) is one modulo N graceful, for every positive integer N.This paper addresses how the concept of one modulo N graceful labelling can be applied in the fields of networking. An overview and new ideas has been proposed here.