Zero-Sum Magic and Null Sets of Planar Graphs

Abstract

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.