Further Results On Super Edge Magic Deficiency Of Unicyclic Graphs

Abstract

A graph \(G\) is called edge-magic if there exists a bijective function \(\phi: V(G) \cup E(G) \rightarrow \{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that \(\phi(x) + \phi(xy) + f\phi(y) = c(\phi)\) is a constant for every edge \(xy \in E(G)\), called the valence of \(\phi\). A graph \(G\) is said to be super edge-magic if \(\phi(V(G)) = \{1, 2, \ldots, |V(G)|\}\). The super edge-magic deficiency, denoted by \(\mu_s(G)\), is the minimum nonnegative integer \(n\) such that \(G \cup nK_1\) has a super edge-magic labeling, if such integer does not exist we define \(\mu_s(G)\) to be \(+\infty\). In this paper, we study the super edge-magic deficiency of some families of unicyclic graphs.