Application of the Combinatorial Nullstellensatz to magic-type graph labelings

Abstract

Let $G=(V,E)$ be a simple graph, and let $k\geq 2$ be an integer. For an edge labeling $h:E(G)\to \mathbb{Z}_{k} \backslash \{0\}$, define the induced vertex label by \[ h^+(v)=\sum_{e \ni v} h(e) \pmod{k}. \] For $t\in \mathbb Z_k$, we say that $G$ is \emph{$t$-sum $\mathbb Z_k$-magic} if there exists such a labeling $h$ satisfying \[ h^+(v)=t \qquad\text{for all }v\in V. \] We say that $G$ is \emph{$\mathbb Z_k$-magic} if $G$ is $t$-sum $\mathbb Z_k$-magic for some $t\in \mathbb Z_k$. Similarly, if there exists an edge labeling $h: E(G) \to \mathbb{Z}_{k} \backslash \{0\}$ such that the induced vertex labeling $h^+(v)=\sum_{e\ni v} h(e)$ (mod $k$) is injective, then $G$ is called \emph{$\mathbb{Z}_{k}$-antimagic}. In this paper, we use the Combinatorial Nullstellensatz to analyze these two types of magic graph labelings.

Disclosure

“rticular, one may ask for conditions under which the Combinatorial Nullstellensatz recovers bounds closer to those expected from explicit Zp -antimagic constructions. 6. Tool and computational resource disclosure ChatGPT was used to help draw figures and proofread the manuscript for grammar, spelling, and punctuation. 7. Acknowledgments The authors gratefully acknowledge the welcoming atmosphere of the 57th Sou”

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Proofreading, grammar, or spelling

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2026-08-23T19:02:03.422133Z

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