Zero-Sum Magic and Null Sets of Planar Graphs

For any \( h \in \mathbb{N} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h \setminus \{0\} \) such that the induced vertex labeling \( l^+: V(G) \to \mathbb{Z}_h \), defined by

\[ l^+(v) = \sum_{uv \in E(G)} l(uv), \]

is a constant map. When this constant is \( 0 \), we call \( G \) a zero-sum \( h \)-magic graph. The null set of \( G \) is the set of all natural numbers \( h \in \mathbb{N} \) for which \( G \) admits a zero-sum \( h \)-magic labeling. A graph \( G \) is said to be uniformly null if every magic labeling of \( G \) induces a zero sum. In this paper, we will identify the null sets of certain planar graphs such as wheels and fans.