Super Fibonacci Graceful Labeling

Abstract

Abstract: A Smarandache-Fibonacci Triple is a sequence S(n), n ≥ 0 such that S(n) = S(n−1)+S(n−2), where S(n) is the Smarandache function for integers n ≥ 0. Certainly, it is a generalization of Fibonacci sequence. A Fibonacci graceful labeling and a super Fibonacci graceful labeling on graphs were introduced by Kathiresan and Amutha in 2006. Generally, let G be a (p,q)-graph and {S(n)|n ≥ 0} a Smarandache-Fibonacci Triple. An bijection f: V (G) → {S(0), S(1), S(2),..., S(q)} is said to be a super Smarandache-Fibonacci graceful graph if the induced edge labeling f ∗ (uv) = |f(u) − f(v) | is a bijection onto the set {S(1), S(2),..., S(q)}. Particularly, if S(n), n ≥ 0 is just the Fibonacci sequence Fi, i ≥ 0, such a graph is called a super Fibonacci graceful graph. In this paper, we construct new types of graphs namely Fn ⊕ K + 1,m, Cn ⊕ Pm, K1,n ⊘ K1,2, Fn ⊕ Pm and Cn ⊕ K1,m and we prove that these graphs are super Fibonacci graceful graphs