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 = (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 A, eE(v)ϕ(e) = a for every v 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 -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.

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