The k-maximal hypergraph of commutative rings

Abstract

Let R be a commutative ring with identity, \(k\ge 2\) a fixed integer and \(\mathcal {I}(R,k)\) be the set of all k-maximal elements in R. The k-maximal hypergraph associated with R, denoted by \(\mathcal {H}^k(R)\), is a hypergraph with the vertex set \(\mathcal {I}(R, k)\) and for distinct elements \(a_1, a_2,\ldots , a_k\) in \(\mathcal {I}(R, k)\) the set \(\{a_1, a_2,\ldots , a_k\}\) is an edge of \(\mathcal {H}^k(R)\) if and only if \(\sum \nolimits _{i=1}^{k} Ra_{i}=R\) and for all \(1\le j\le k\). In this paper, the connectedness, diameter and girth of \(\mathcal {H}^k(R)\) are studied. Moreover, the regularity and coloring of \(\mathcal {H}^k(R)\) are investigated. Among other things, we characterize all finite commutative rings R for which the k-maximal hypergraph \(\mathcal {H}^k(R)\) is outerplanar and planar.

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Acknowledgements

The work is supported by the SERB-EEQ project (EEQ/2016/000367) of Department of Science and Technology, Government of India for the first author. Also the work reported here is supported by the INSPIRE programme (IF 160175) awarded to the second author by the Deparment of Science and Technology, Government of India.

Funding

The work reported here is supported by the INSPIRE programme (IF160175) of Department of Science and Technology, Government of India for the second author.

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Correspondence to V. C. Amritha.

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Selvakumar, K., Amritha, V.C. The k-maximal hypergraph of commutative rings. Beitr Algebra Geom 61, 747–757 (2020). https://doi.org/10.1007/s13366-020-00505-8

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Keywords

  • Co-maximal graph
  • k-maximal hypergraph
  • Connectedness
  • Girth
  • Planar graph

Mathematics Subject Classification

  • 05C12
  • 05C25
  • 05C69
  • 13A15