On the Super Edge-Magic Deficiency of a Star Forest

Abstract

Let \(G = (V, E)\) be a finite, simple, and undirected graph of order \(p\) and size \(q\). A super edge-magic total labeling of a graph \(G\) is a bijection \(\lambda: V(G) \cup E(G) \rightarrow \{1, 2, \ldots, p + q\}\), where vertices are labeled with \(1, 2, \ldots, p\) and there exists a constant \(t\) such that \(f(x) + f(xy) +f(y) = t\), for every edge \(xy \in E(G)\). The super edge-magic deficiency of a graph \(G\), denoted by \(\mu_s(G)\), is the minimum nonnegative integer \(n\) such that \(G \cup nK_1\) has a super edge-magic total labeling, or \(\infty\) if no such \(n\) exists. In this paper, we investigate the super edge-magic deficiency of a forest consisting of stars.