Powers of Binomial Edge Ideals With Quadratic Gröbner Bases

Abstract

We study powers of binomial edge ideals associated with closed and block graphs.

On the Equality of Symbolic and Ordinary Powers of Binomial Edge Ideals

“…For block graphs with Cohen-Macaulay binomial edge ideals, those one with equality of ordinary and symbolic powers are characterized in [5,Theorem 4.1]. We give a partial generalization of that result.…”

“…Although the equality of symbolic and ordinary powers of JG for a block graph G is equivalent to closedness of G when JG is Cohen-Macaulay, it turns out in Theorem 3.11 that when G is a generalized caterpillar, that equality holds exactly when G is weakly closed. Moreover, either G is a block graph with Cohen-Macaulay binomial edge ideal or G is a generalized caterpillar graph, then equality of symbolic and ordinary powers is equivalent to G being net-free, see [5,Theorem 4.1] and Theorem 3.11.…”

Regularity of powers of (parity) binomial edge ideals

“…To show (8), we first notice that by Lemma 4.1, we may suppose that the length of the path added to H is 1. Let e ′ be the only edge of this path.…”

“…As one can observe, for both binomial edge ideals and parity binomial edge ideals, results on the regularity of their powers are not abundant. The work in [8] and [17] are perhaps the only ones along this research line that we can find so far. That is the reason why we want to push forward a little bit.…”

“…Only recently, Jayanthan et al in [17] provided the exact formulas for the regularity of powers of binomial edge ideals of several simple graphs such as cycle graphs, star graphs and balloon graphs. And Ene et al in [8] studied the same problems for connected closed graphs.…”