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.