On the essential graph of a poset

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Document Type : Original paper

Authors

1 Department of Mathematics, Jundi-Shapur University of Technology, Dezful, Iran

2 Department of Physics, Technical and Vocational University (TVU), Tehran, Iran

Abstract

Let $(P, \leq)$ be an atomic partially ordered set (briefly, a poset) with a minimum element $0$, and let $\mathcal{I}(P)$ be the set of all nontrivial ideals of $ P $. The essential graph of $P$, denoted by $G_e(P)$, is an undirected, simple graph with the vertex set $\mathcal{I}(P)$ and two distinct vertices $I, J \in \mathcal{I}(P) $ are adjacent in $G_e(P)$ if and only if $ I\cup J $ is an essential ideal of $P$. We study the connections between the graph-theoretic properties of this graph and the algebraic properties of a poset. We prove that $G_e(P)$ is connected with diameter at most three. Furthermore, all posets are characterized based on the diameters of their essential graphs. Also, all posets with planar $G_e(P)$ are classified. Among other results, the clique number and chromatic number of $G_e(P)$ are determined.

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References

[1] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer Publishing Company, Incorporated, New York, 2008.
[2] Y. Civan, Upper maximal graphs of posets, Order 30 (2013), 677–688. https://doi.org/10.1007/s11083-012-9270-4
[3] R. Halaš and M. Jukl, On Beck’s coloring of posets, Discrete Math. 309 (2009), no. 13, 4584–4589. https://doi.org/10.1016/j.disc.2009.02.024
[4] V. Joshi, Zero divisor graph of a poset with respect to an ideal, Order 29 (2012), 499–506. https://doi.org/10.1007/s11083-011-9216-2
[5] J.D. LaGrange and K.A. Roy, Poset graphs and the lattice of graph annihilators, Discrete Math. 313 (2013), no. 10, 1053–1062.  https://doi.org/10.1016/j.disc.2013.02.004
[6] M.J. Nikmehr, R. Nikandish, and M. Bakhtyiari, On the essential graph of a commutative ring, J. Algebra Appl. 16 (2017), no. 7, Article ID: 1750132. https://doi.org/10.1142/S0219498817501328
[7] S.K. Nimbhorkar, M.P. Wasadikar, and L. DeMeyer, Coloring of semilattices, Ars Combin. 12 (2007), 97–104.
[8] S. Roman, Lattices and Ordered Sets, Springer Science & Business Media, New York, 2008.
[9] S. Rudeanu, Sets and Ordered Structures, Bentham Science Publishers, 2012.
[10] D.B. West, Introduction to Graph Theory, Prentice hall Upper Saddle River, 2001.
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 29 June 2025

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