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Depth, Stanley depth, and regularity of ideals associated to graphs

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Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rm sdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rm depth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).

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Seyed Fakhari, S.A. Depth, Stanley depth, and regularity of ideals associated to graphs. Arch. Math. 107, 461–471 (2016). https://doi.org/10.1007/s00013-016-0965-4

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