A simple-intersection graph of a ring approach to solving coloring optimization problems

Document Type : Original paper

Authors

Department of Basic Sciences, Princess Sumaya University for Technology, Amman, Jordan

Abstract

In this paper, we introduce a modified version of the simple-intersection graph for semisimple rings, applied to a ring $R$ with unity. The findings from this modified version are subsequently utilized to solve several coloring optimization problems.  We demonstrate how the clique number of the simple-intersection graph can be used to determine the maximum number  of possibilities that can be selected from a set of $n$ colors without replacement or order, subject to the constraint that  any pair shares only one common color. We also show how the domination number can be used to determine the  minimum number of possibilities that can be selected, such that any other possibility shares one color with  at least one of the selected possibilities, is $n-1$.

Keywords

Main Subjects

References

[1] S. Akbari and R. Nikandish, Some results on the intersection graph of ideals of matrix algebras, Linear Multilinear Algebra 62 (2014), no. 2, 195–206.  https://doi.org/10.1080/03081087.2013.769101
[2] S. Akbari, R. Nikandish, and M.J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl. 12 (2013), no. 4, Article ID: 1250200.  https://doi.org/10.1142/S0219498812502003
[3] T. Alraqad, H. Saber, and R. Abu-Dawwas, Intersection graphs of graded ideals of graded rings, AIMS Math. 6 (2021), 10355–10368.  https://doi.org/10.3934/math.2021600
[4] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434–447. https://doi.org/10.1006/jabr.1998.7840
[5] N. Ashrafi, H.R. Maimani, M.R. Pournaki, and S. Yassemi, Unit graphs associated with rings, Comm Algebra 38 (2010), no. 8, 2851–2871.  https://doi.org/10.1080/00927870903095574
[6] J. A. Bondy and U.S.R. Murty, Graph Theory, Springer-Verlag, London, 2011.
[7] I. Chakrabarty, S. Ghosh, T.K. Mukherjee, and M.K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (2009), no. 17, 5381–5392.  https://doi.org/10.1016/j.disc.2008.11.034
[8] I. Chakrabarty and J.V. Kureethara, A survey on the intersection graphs of ideals of rings, Commun. Comb. Optim. 7 (2022), no. 2, 121–167.  https://doi.org/10.22049/cco.2021.26990.1176
[9] P.M. Cohn, Introduction to Ring Theory, Springer Science & Business Media, London, 2012.
[10] P. Devi and R. Rajkumar, On the complement of the intersection graph of subgroups of a group, Commun. Comb. Optim., 10 (2025), no. 1, 57–68. https://doi.org/10.22049/cco.2023.28198.1476
[11] S.H. Jafari and N. Jafari Rad, Planarity of intersection graphs of ideals of rings, Int. Electron. J. Algebra 8 (2010), 161–166.
[12] N. Jafari Rad, S.H. Jafari, and S. Ghosh, On the intersection graphs of ideals of direct product of rings, Discuss. Math. Gen. Algebra Appl. 34 (2014), no. 2, 191–201.  http://dx.doi.org/10.7151/dmgaa.1224
[13] A.V. Kelarev, On undirected cayley graphs, Australas. J. Combin. 25 (2002), 73–78.
[14] L.A. Mahdavi and Y. Talebi, Co-intersection graph of submodules of a module, Algebra Discrete Math. 21 (2016), no. 1, 128–143.
[15] H.R. Maimani, M. Salimi, A. Sattari, and S. Yassemi, Comaximal graph of commutative rings, J. Algebra 319 (2008), no. 4, 1801–1808.  https://doi.org/10.1016/j.jalgebra.2007.02.003
[16] J. Matczuk and A. Majidinya, Sum-essential graphs of modules, J. Algebra Appl. 20 (2021), no. 11, Article ID: 2150211. https://doi.org/10.1142/S021949882150211X
[17] J. Matczuk, M. Nowakowska, and E.R. Puczyłowski, Intersection graphs of modules and rings, J. Algebra Appl. 17 (2018), no. 07, Article ID: 1850131.  https://doi.org/10.1142/S0219498818501311
[18] F. Moh’d and M. Ahmed, Simple-intersection graphs of rings, AIMS Math. 8 (2023), no. 1, 1040–1054.  https://doi.org/10.3934/math.2023051
[19] E. A. Osba, The intersection graph for finite commutative principal ideal rings, Acta Math. Acad. Paedagog. Nyiregyhaziensis 32 (2016), no. 1, 15–22.
[20] Z.S. Pucanović and Z.Z. Petrović, Toroidality of intersection graphs of ideals of commutative rings, Graphs Combin. 30 (2014), 707–716.  https://doi.org/10.1007/s00373-013-1292-1
Communications in Combinatorics and Optimization
Volume 10, Issue 2
June 2025
Pages 423-442

Files

Share

How to cite

Statistics