Depth and Stanley depth of the edge ideals of the strong product of some graphs

Year 2021, , 92 - 109, 04.02.2021

Zahid Iqbal , Muhammad Ishaq Muhammad Ahsan Binyamin

https://doi.org/10.15672/hujms.638033

Cited By: 4

Abstract

In this paper, we study depth and Stanley depth of the edge ideals and quotient rings of the edge ideals, associated with classes of graphs obtained by the strong product of two graphs. We consider the cases when either both graphs are arbitrary paths or one is an arbitrary path and the other is an arbitrary cycle. We give exact formula for values of depth and Stanley depth for some subclasses. We also give some sharp upper bounds for depth and Stanley depth in the general cases.

Keywords

Depth, Stanley depth, Stanley decomposition, monomial ideal, edge ideal, strong product of graphs

References

• [1] C. Bir, D.M. Howard, M.T. Keller, W.T. Trotter and S.J. Young, Interval partitions and Stanley depth, J. Combin. Theory Ser. A, 117, 475–482, 2010.
• [2] M. Cimpoeaş, Several inequalities regarding Stanley depth, Romanian Journal of Math. and Computer Science, 2, 28–40, 2012.
• [3] M. Cimpoeaş, Stanley depth of squarefree Veronese ideals, An. St. Univ. Ovidius Constanta, 21 (3), 67–71, 2013.
• [4] M. Cimpoeaş, On the Stanley depth of edge ideals of line and cyclic graphs, Romanian Journal of Math. and Computer Science, 5 (1), 70–75, 2015.
• [5] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, available at http://cocoa.dima.unige.it.
• [6] A.M. Duval, B. Goeckner, C.J. Klivans and J.L. Martine, A non-partitionable Cohen- Macaulay simplicial complex, Adv. Math. 299, 381–395, 2016.
• [7] S.A.S. Fakhari, On the Stanley Depth of Powers of Monomial Ideals, Mathematics, 7, 607, 2019.
• [8] L. Fouli and S. Morey, A lower bound for depths of powers of edge ideals, J. Algebraic Combin. 42 (3), 829–848, 2015.
• [9] R. Hammack, W. Imrich and S. Klavar, Handbook of Product Graphs, Second Edition, CRC Press, Boca Raton, FL, 2011.
• [10] J. Herzog, A survey on Stanley depth, In Monomial ideals, computations and applications, Lecture Notes in Math. 2083, Springer, Heidelberg, 3–45, 2013.
• [11] J. Herzog, M. Vladoiu and X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (9), 3151–3169, 2009.
• [12] Z. Iqbal and M. Ishaq, Depth and Stanley depth of edge ideals associated to some line graphs, AIMS Mathematics, 4 (3), 686–698, 2019.
• [13] Z. Iqbal and M. Ishaq, Depth and Stanley depth of edge ideals of powers of paths and cycles, An. Şt. Univ. Ovidius Constana, 27 (3), 113–135, 2019.
• [14] Z. Iqbal, M. Ishaq and M. Aamir, Depth and Stanley depth of edge ideals of square paths and square cycles, Comm. Algebra, 46 (3), 1188–1198, 2018.
• [15] M. Ishaq, Upper bounds for the Stanley depth, Comm. Algebra, 40 (1), 87–97, 2012.
• [16] M. Ishaq, Values and bounds for the Stanley depth, Carpathian J. Math. 27 (2), 217–224, 2011.
• [17] M. Ishaq and M.I. Qureshi, Upper and lower bounds for the Stanley depth of certain classes of monomial ideals and their residue class rings, Comm. Algebra, 41 (3), 1107–1116, 2013.
• [18] M.T. Keller and S.J. Young, Combinatorial reductions for the Stanley depth of I and S/I, Electron. J. Comb. 24 (3), #P3.48, 2017.
• [19] M.T. Keller, Y. Shen, N. Streib and S.J. Young, On the Stanley depth of squarefree veronese ideals, J. Algebraic Combin. 33 (2), 313–324, 2011.
• [20] S. Morey, Depths of powers of the edge ideal of a tree, Comm. Algebra, 38 (11), 4042–4055, 2010.
• [21] R. Okazaki, A lower bound of Stanley depth of monomial ideals, J. Commut. Algebra, 3 (1), 83–88, 2011.
• [22] M.R. Pournaki, S.A.S. Fakhari and S. Yassemi, Stanley depth of powers of the edge ideals of a forest, Proc. Amer. Math. Soc. 141 (10), 3327–3336, 2013.
• [23] M.R. Pournaki, S.A.S. Fakhari, M. Tousi and S. Yassemi, What is . . . Stanley depth? Not. Am. Math. Soc. 56, 1106–1108, 2009.
• [24] A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38 (2), 773–784, 2010.
• [25] G. Rinaldo, An algorithm to compute the Stanley depth of monomial ideals, Le Matematiche, LXIII(ii), 243–256, 2008.
• [26] R.P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (2), 175–193, 1982.
• [27] A. Stefan, Stanley depth of powers of path ideal, http://arxiv.org/pdf/1409.6072.pdf.
• [28] R.H. Villarreal, Monomial Algebras in:Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 238, 2011.