# Genus and book thickness of reduced cozero-divisor graphs of commutative rings

## DOI:

https://doi.org/10.33044/revuma.3906## Abstract

For a commutative ring $R$ with identity, let $\langle a\rangle$ be the principal ideal generated by $a\in R$. Let $\Omega(R)^*$ be the set of all nonzero proper principal ideals of $R$. The reduced cozero-divisor graph $\Gamma_r(R)$ of $R$ is the simple undirected graph whose vertex set is $\Omega(R)^*$ and such that two distinct vertices $\langle a\rangle$ and $\langle b\rangle$ in $\Omega(R)^\ast$ are adjacent if and only if $\langle a \rangle\nsubseteq\langle b\rangle$ and $\langle b\rangle\nsubseteq\langle a\rangle$. In this article, we study certain properties of embeddings of the reduced cozero-divisor graph of commutative rings. More specifically, we characterize all Artinian nonlocal rings whose reduced cozero-divisor graph has genus two. Also we find the book thickness of the reduced cozero-divisor graphs which have genus at most one.

### Downloads

## References

M. Afkhami and K. Khashyarmanesh, The cozero-divisor graph of a commutative ring, *Southeast Asian Bull. Math.* **35** no. 5 (2011), 753–762. MR Zbl

D. F. Anderson, T. Asir, A. Badawi, and T. Tamizh Chelvam, *Graphs from rings*, Springer, Cham, 2021. DOI MR Zbl

D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, *J. Algebra* **217** no. 2 (1999), 434–447. DOI MR Zbl

D. Archdeacon, Topological graph theory: a survey, *Congr. Numer.* **115** (1996), 5–54. MR Zbl

M. F. Atiyah and I. G. Macdonald, *Introduction to commutative algebra*, Addison-Wesley, Reading, Mass., 1969. MR Zbl

I. Beck, Coloring of commutative rings, *J. Algebra* **116** no. 1 (1988), 208–226. DOI MR Zbl

M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, *J. Algebra Appl.* **10** no. 4 (2011), 727–739. DOI MR Zbl

F. Bernhart and P. C. Kainen, The book thickness of a graph, *J. Combin. Theory Ser. B* **27** no. 3 (1979), 320–331. DOI MR Zbl

J. A. Bondy and U. S. R. Murty, *Graph theory with applications*, American Elsevier, New York, 1976. MR Zbl

I. Chakrabarty, S. Ghosh, T. K. Mukherjee, and M. K. Sen, Intersection graphs of ideals of rings, *Discrete Math.* **309** no. 17 (2009), 5381–5392. DOI MR Zbl

G. Chartrand and F. Harary, Planar permutation graphs, *Ann. Inst. H. Poincaré Sect. B (N.S.)* **3** (1967), 433–438. MR Zbl

E. Jesili, K. Selvakumar, and T. Tamizh Chelvam, On the genus of reduced cozero-divisor graph of commutative rings, *Soft Comput.* **27** (2023), 657–666. DOI

S. Kavitha and R. Kala, On the genus of graphs from commutative rings, *AKCE Int. J. Graphs Comb.* **14** no. 1 (2017), 27–34. DOI MR Zbl

B. Mohar and C. Thomassen, *Graphs on surfaces*, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2001. MR Zbl

K. Selvakumar, M. Subajini, and M. J. Nikmehr, Finite commutative ring with genus two essential graph, *J. Algebra Appl.* **17** no. 7 (2018), Paper No. 1850121, 11 pp. DOI MR Zbl

F. Shaveisi and R. Nikandish, The nil-graph of ideals of a commutative ring, *Bull. Malays. Math. Sci. Soc.* **39** suppl. 1 (2016), S3–S11. DOI MR Zbl

T. Tamizh Chelvam and T. Asir, On the genus of the total graph of a commutative ring, *Comm. Algebra* **41** no. 1 (2013), 142–153. DOI MR Zbl

H.-J. Wang, Zero-divisor graphs of genus one, *J. Algebra* **304** no. 2 (2006), 666–678. DOI MR Zbl

A. T. White, *Graphs, groups and surfaces*, North-Holland Mathematics Studies, No. 8, North-Holland, Amsterdam-London; American Elsevier, New York, 1973. MR Zbl

C. Wickham, Rings whose zero-divisor graphs have positive genus, *J. Algebra* **321** no. 2 (2009), 377–383. DOI MR Zbl

A. Wilkens, J. Cain, and L. Mathewson, Reduced cozero-divisor graphs of commutative rings, *Int. J. Algebra* **5** no. 17-20 (2011), 935–950. MR Zbl

M. Ye and T. Wu, Co-maximal ideal graphs of commutative rings, *J. Algebra Appl.* **11** no. 6 (2012), Paper No. 1250114, 14 pp. DOI MR Zbl

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2024 Edward Jesili, Krishnan Selvakumar, Thirugnanam Tamizh Chelvam

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.