Let G be a graph with n vertices and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over a field K. Assume that J(G) is the cover ideal of G and J(G) (k) is its k-th symbolic power. We prove that if G is a very well-covered graph such that J(G) has linear resolution, then J(G) (k) has linear resolution, for every integer k ≥ 1. We also prove that for a every very well-covered graph G, the depth of symbolic powers of J(G) forms a non-increasing sequence. Finally, we determine a linear upper bound for the regularity of powers of cover ideal of bipartite graph.

Abstract. Let G = W (C n ) be a whiskered cycle graph with edge ideal I = I(G). We prove that for every s ≥ 1, the equality reg(I s ) = 2s + ⌈ n−1 2 ⌉ − 1 holds.

Abstract. Let K be a field and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over the field K. Let G be a forest with p connected components G 1 , . . . , G p and let I = I(G) be its edge ideal in S. Suppose that d i is the diameter of G i , 1 ≤ i ≤ p, and consider d = max {d i | 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max+ p − 1, p is a lower bound for depth(S/I t ). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/I t ). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.

No abstract

Let G be a graph with edge ideal I(G). We recall the notions of min-match {K2,C5} (G) and ind-match {K2,C5} (G) from [23]. We show thatMoreover, we show thatwhere ord-match(G) denotes the ordered matching number of G. Finally, we construct infinitely many connected graphs which satisfy the following strict inequalities:2s + ind-match(G) − 1 < reg(I(G) s ) < 2s + cochord(G) − 1. This gives a positive answer to a question asked in [15].