2026/07/22 by Parikshit Chalise, Richard M. Low
Mathematics · #math.CO
Let G=(V,E) be a simple graph, and let k≥ 2 be an integer. For an edge labeling h:E(G)→ ℤk \backslash \0\, define the induced vertex label by h+(v)=∑e \ni v h(e) \pmodk. For t∈ \mathbb Zk, we say that G is t-sum \mathbb Zk-magic if there exists such a labeling h satisfying h+(v)=t \qquadfor all v∈ V. We say that G is \mathbb Zk-magic if G is t-sum \mathbb Zk-magic for some t∈ \mathbb Zk. Similarly, if there exists an edge labeling h: E(G) → ℤk \backslash \0\ such that the induced vertex labeling h+(v)=∑e\ni v h(e) (mod k) is injective, then G is called \emphℤk-antimagic. In this paper, we use the Combinatorial Nullstellensatz to analyze these two types of magic graph labelings.