Open Access

## A. Kumar, V. Kumar, K. Kumar, P. Gupta and Y. Khandelwal

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After a long period of scramble over analysis and investigations, the notion of graceful labeling came into existence. A mapping $f$ for a graph $G=(R,S)$ is said to be graceful if there exists a bijective mapping $f:R\left(G\right)\to N\cup \{0\}$ such that each edge has an induced label $\omega \left(f,\ R\left(G\right)\right)=\left\{\left|f\left(u\right)-f\left(v\right)\right|:u,\ v\ \epsilon \ R\left(G\right)\right\}$ and the resulting edge labels are distinct. In this paper, we introduce a new type of graph labeling for a graph $G\, =\, (R,\, S\, )$ which we call $G$-graceful labeling. The $G$-graceful labeling for the graph $G\, =\, (R,\, S)$ with $r$ vertices and $s$edges is an injective function $\rho \, :\, R(G)\, \to \, \{ 0,\, 1,\, 2,\, 3,\, …,\, t-1\}$such that the induced function $\rho ^{*\, \, } :S(G)\, \to \, N$ is given by$\rho ^{*} (r,\, s)\, =\{ \rho ^{*} (r)\, +\, \rho ^{*} (s)\}$,the resulting edge label are distinct. In this paper, we also prove that the graphs: path, ladder graph, flower graph, complete bipartite graph and star graph admit $G$-graceful labeling.

Keywords:
G-graceful graph, path, ladder graph, flower graph, complete bipartitegraph, star graph.