On magic type labelings of zero-divisor graphs

Abstract

In this article, we investigate magic type labelings of zero-divisor graphs. In particular, we turn our attention to semi-magic, magic, and super-magic labelings. We are able to construct infinitely many rings which admit these magic type labelings as well as infinitely many rings which do not have these magic type labeling. We further proceed to classify the magic type labeling properties for all of the rings which have zero-divisor graphs with up to 14 vertices. We then conclude with some conjectures about how these patterns may extend for larger zero-divisor graphs.

Keywords

• Zero-divisor graph
• magic labeling
• semi-magic
• super-magic

References

1. D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra, 159 (1993), 500-514.
2. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434-447.
3. I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988), 208-226.
4. C. P. Mooney, On gracefully and harmoniously labeling zero-divisor graphs, in Rings, Monoids and Module Theory, Springer Proc. Math. Stat., Springer, Singapore, 382 (2021), 239-260.
5. S. P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete Math., 307(9-10) (2007), 1155-1166.
6. S. P. Redmond, Corrigendum to: “On zero-divisor graphs of small finite commutative rings” [Discrete Math. 307(9-10) (2007), 1155–1166], Discrete Math., 307(21) (2007), 2449-2452.
7. J. Sedlacek, On magic graphs, Math. Slovaca, 26(4) (1976), 329-335.
8. B. M. Stewart, Magic graphs, Canadian J. Math., 18 (1966), 1031-1059.

9. B. M. Stewart, Supermagic complete graphs, Canadian J. Math., 19 (1967), 427-438.