On the Structures of $V_4$-Magic and ${\mathcal{Z}}_4$-Magic Graphs

Abstract

This paper settles in the negative the following open question: Are $V_4$-magic graphs necessarily ${\mathbb{Z}}_4$-magic? For an abelian group $A$, we examine the properties of $A$-magic labelings with constant weight $0$, called $zero-sumA-magic$, and utilize well-known results on edge-colorings in order to construct (from $3$-regular graphs) infinite families that are $V_4$-magic but not ${\mathbb{Z}}_4$-magic. Noting that our arguments lead to connected graphs of order $2n$ for all $n\ge 11$ that are $V_4$-magic and not ${\mathbb{Z}}_4$-magic, we conclude the paper by investigating the zero-sum integer-magic spectra of graphs, including Cartesian products, and give a conjecture about the zero-sum integer-magic spectra of $3$-regular graphs.