On the (Super) Edge-Magic Deficiency of Chain Graphs

A graph \( G \) of order \( |V(G)| \) and size \( |E(G)| \) is called edge-magic if there exists a bijection \( f : V(G) \cup E(G) \to \{1, 2, 3, \dots, |V(G)| + |E(G)|\} \) such that \( f(x) + f(xy) + f(y) \) is a constant for every edge \( xy \in E(G) \). An edge-magic graph \( G \) is said to be super if \( f(V(G)) = \{1, 2, 3, \dots, |V(G)|\} \). Furthermore, the edge-magic deficiency of a graph \( G \), denoted \( \mu(G) \), is defined as the minimum nonnegative integer \( n \) such that \( G \cup nK_1 \) is edge-magic. Similarly, the \emph{super edge-magic deficiency} of a graph \( G \), denoted \( \mu_s(G) \), is either the minimum nonnegative integer \( n \) such that \( G \cup nK_1 \) is super edge-magic or \( +\infty \) if there exists no such integer \( n \). In this paper, we investigate the (super) edge-magic deficiency of chain graphs. Based on these, we propose some open problems.