On the zero-sum group-magicness of cartesian products

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

doi:10.2140/involve.2019.12.1261

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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

DOI: 10.2140/involve.2019.12.1261

Abstract

Let $G = (V (G), E (G))$ be a graph and let $A = (A, +)$ be an abelian group with identity 0. Then an $A$ -magic labeling of $G$ is a function $ϕ$ from $E (G)$ into $A ∖ {0}$ such that for some $a \in A$ , $\sum_{e \in E (v)} ϕ (e) = a$ for every $v \in V (G)$ , where $E (v)$ is the set of edges incident to $v$ . If $ϕ$ exists such that $a = 0$ , then $G$ is zero-sum $A$ -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 $j^{*} (G)$ to explore the zero-sum integer-magic spectrum (or null space) of $G$ . For certain $G$ , we establish $A (G)$ , the set of nontrivial abelian groups for which $G$ is zero-sum group-magic. Particular attention is given to $A (G)$ 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

Mathematical Subject Classification 2010

Primary: 05C78

Milestones

Received: 14 March 2018

Revised: 27 August 2019

Accepted: 20 September 2019

Published: 25 October 2019

Communicated by Kenneth S. Berenhaut

Authors

Adam Fong
Department of Mathematics Trinity College Hartford, CT United States

John Georges
Department of Mathematics Trinity College Hartford, CT United States

David Mauro
Department of Mathematics Trinity College Hartford, CT United States

Dylan Spagnuolo
Department of Mathematics Trinity College Hartford, CT United States

John Wallace
Department of Mathematics Trinity College Hartford, CT United States

Shufan Wang
Department of Mathematics Trinity College Hartford, CT United States

Kirsti Wash
Department of Mathematics Trinity College Hartford, CT United States

Vol. 12, No. 8, 2019