On the Super Edge-Magic Deficiency of \(2\)-Regular Graphs with Two Components

Abstract

A graph \(G\) is called super edge-magic if there exists a bijective function \(f: V(G) \cup E(G) \rightarrow \{1, 2, \dots, |V(G)| + |E(G)|\}\) such that \(f(V(G)) = \{1, 2, \dots, |V(G)|\}\) and \(f(u) + f(v) + f(uv)\) is a constant for each \(uv \in E(G)\). The super edge-magic deficiency, \(\mu_s(G)\), of a graph \(G\) is defined as the smallest nonnegative integer \(n\) with the property that the graph \(G \cup nK_1\) is super edge-magic, or \(+\infty\) if there exists no such integer \(n\). In this paper, the super edge-magic deficiency of certain 2-regular graphs with two components is computed, which leads us to a conjecture on the super edge-magic deficiency of graphs in this class.