Depth, Stanley depth and regularity of ideals associated to graphs

2016/04/03 by Fakhari, S. A. Seyed
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.00656

Abstract

Let \mathbbK be a field and S=\mathbbK[x₁,…,x_n] be the polynomial ring in n variables over \mathbbK. 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)≥ n-ν_o(G) and \rm sdepth(S/J)≥ n-ν_o(G)-1, where ν_o(G) is the ordered matching number of G. We also prove the inequalities \rm sdepth(J^k)≥ \rm depth(J^k) and \rm sdepth(S/J^k)≥ \rm depth(S/J^k), for every integer k≫ 0, when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality \rm reg(S/I)≤ ν_o(G).

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