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

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 ({{rmsdepth}(J^k)geq {rm depth}(J^k)}) and ({{rm sdepth}(S/J^k)geq {rmdepth}(S/J^k)}), for every integer ({kgg 0}), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg({(S/I)leq nu_{o}(G)}).