Odd triangular graceful labeling on simple graphs

Abstract

In 2001, Devaraj et. al. [1] defined Triangular graceful graphs. We shall define an odd triangular graceful graphs as follows:Let \(V(G)\) and \(E(G)\) denote the vertex set and edge set of the graph \(G\) respectively. Consider an injective function \(f: V(G) \rightarrow\left\{0,1,2, \ldots, T_{2 q-1}\right\}\), where \(q\) is the number of edges of \(G\) and \(T_i\) is the \(i^{\text {th }}\) triangular number. That is \(T_1=1, T_2=3, T_3=6\) etc., and \(T_n=\frac{n(n+1)}{2}\). If the function \(f\) induces the function \(f^*\) on \(E(G)\) such that \(f^*(u v)=|f(u)-f(v)|\) for all edges \((u v) \in E(G)\) with \(\left\{f^*(E(G))\right\}=\left\{T_1, T_3, \ldots, T_{2 q-1}\right\}\), we say that \(f\) is an odd triangular graceful graph and a graph which admits such a labeling is called an odd triangular graceful labeling.