Vol. 12, No. 8, 2019

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On the zero-sum group-magicness of cartesian products

Adam Fong, John Georges, David Mauro, Dylan Spagnuolo, John Wallace, Shufan Wang and Kirsti Wash

Vol. 12 (2019), No. 8, 1261–1278
Abstract

Let $G=\left(V\left(G\right),E\left(G\right)\right)$ be a graph and let $\mathbb{A}=\left(A,+\right)$ be an abelian group with identity 0. Then an $\mathbb{A}$-magic labeling of $G$ is a function $\varphi$ from $E\left(G\right)$ into $A\setminus \left\{0\right\}$ such that for some $a\in A$, ${\sum }_{e\in E\left(v\right)}\varphi \left(e\right)=a$ for every $v\in V\left(G\right)$, where $E\left(v\right)$ is the set of edges incident to $v$. If $\varphi$ exists such that $a=0$, then $G$ is zero-sum $\mathbb{A}$-magic. Let $G$ be the cartesian product of two or more graphs. We establish that $G$ is zero-sum $ℤ$-magic and we introduce a graph invariant ${j}^{\ast }\left(G\right)$ to explore the zero-sum integer-magic spectrum (or null space) of $G$. For certain $G$, we establish $\mathsc{A}\left(G\right)$, the set of nontrivial abelian groups for which $G$ is zero-sum group-magic. Particular attention is given to $\mathsc{A}\left(G\right)$ for regular $G$, odd/even $G$, and $G$ isomorphic to a product of paths.

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Keywords
cartesian product of graphs, grid graph, magic labeling, group-magic labeling, zero-sum group-magic labeling, zero-sum integer-magic spectrum
Primary: 05C78