\(C_m\)-\(E\)-super magic graceful labeling of graphs

Abstract

A simple graph $G$ admits an $H$-covering if every edge in $E(G)$ belongs to a subgraph of $G$ isomorphic to $H$. The graph $G$ is said to be $H$-magic if there exists a total labeling $f: V(G) \cup E(G) \rightarrow\{1,2, \ldots, p+q\}$ such that for every subgraph $H^{\prime}$ of $G$ isomorphic to $H, \sum_{v \in V\left(H^{\prime}\right)} f(v)+\sum_{e \in E\left(H^{\prime}\right)} f(e)=M$ for some positive integer $M$. An $H$-E-super magic graceful labeling ( $H$ - $E$-SMGL) is a bijection $f: V(G) \cup E(G) \rightarrow\{1,2, \ldots, p+q\}$ with the property $f(E(G))=\{1,2, \ldots, q\}$ such that $\sum_{v \in V\left(H^{\prime}\right)} f(v)-\sum_{e \in E\left(H^{\prime}\right)} f(e)=M$ for some positive integer $M$. In this paper, we introduce $H-E$-SMGL and study $C_m-E$-super magic graceful labeling of generalized book graph.