On The Super Edge-Magic Deficiency of Graphs

Abstract

A $(p,q)$ graph $G$ is called edge-magic if there exists a bijective function $f:V(G)\cup E(G)\to \{1,2,\dots ,p+q\}$ such that $f(u)+f(v)+f(uv)=k$ is a constant for any edge $uv\in E(G)$. Moreover, $G$ is said to be super edge-magic if $f(V(G))=\{1,2,\dots ,p\}$. The question studied in this paper is for which graphs it is possible to add a finite number of isolated vertices so that the resulting graph is super edge-magic. If it is possible for a given graph $G$, then we say that the minimum such number of isolated vertices is the super edge-magic deficiency, ${\mu }_s(G)$ of $G$; otherwise we define it to be $+\mathrm{\infty }$.