H-E-Supermagic labelings of some families of graphs

Abstract

Let $G$ be a simple graph with $p$ vertices and $q$ edges. A simple graph $G$ admits an H-covering 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 bijection   $f: V(G) \cup E(G) \rightarrow \{ 1,2 , \ldots, p + q \}$ such that for every subgraph $H^{'}$ of $G$ isomorphic to $H$, $ \sum\limits_{v \in V(H^{'})} f(v) + \sum\limits_{e \in E(H^{'})} f(e) = M$ for some positive integer $M$. $G$ is said to be $H$-$E$-Supermagic if $f(E(G)) = \{ 1,2 , \ldots, q\}$. In this paper we study $H$-$E$-Supermagic labelings of fans, triangle ladders, graphs obtained by joining a star $K_{1,n}$ with one isolated vertex, books and grids.