Further Results On Super Edge Magic Deficiency Of Unicyclic Graphs

Abstract

A graph $G$ is called edge-magic if there exists a bijective function $\varphi :V(G)\cup E(G)\to \{1,2,\dots ,|V(G)|+|E(G)|\}$ such that $\varphi (x)+\varphi (xy)+f\varphi (y)=c(\varphi )$ is a constant for every edge $xy\in E(G)$, called the valence of $\varphi$. A graph $G$ is said to be super edge-magic if $\varphi (V(G))=\{1,2,\dots ,|V(G)|\}$. The super edge-magic deficiency, denoted by ${\mu }_s(G)$, is the minimum nonnegative integer $n$ such that $G\cup n{K}_1$ has a super edge-magic labeling, if such integer does not exist we define ${\mu }_s(G)$ to be $+\mathrm{\infty }$. In this paper, we study the super edge-magic deficiency of some families of unicyclic graphs.