Graceful labeling on a new family of graphs

Abstract

Abstract:  Let $G$  be  a graph with $V\left(G\right)$ as the vertex set and $q$ edges. Let $u,v$ be in $V\left(G\right)$ and define an injective function $f$ from $V\left(G\right)$ to $\left\{0,1,2,\ldots ,q\right\}$ such that the resulting edges are also distinct and each edge is assigned the value $\left|f\left(u\right)-f\left(v\right)\right|.$  This function $f$ is called the graceful labelling of $G$. A graph which accepts a graceful labelling is called a graceful graph. In this paper, we prove that the graphs obtained by the join of two complete bipartite graphs and the join of two ladder graphs by a path of arbitrary length are graceful.