Lovász–Saks–Schrijver ideals and parity binomial edge ideals of graphs

Regularity of powers of (parity) binomial edge ideals

“…We claim that reg(S/I G ) = n − 2. With this, since I G is an almost complete intersection of height n − 1 by the proof of [24,Theorem 3.8], it follows from the first case of Corollary 3.8 that reg(S/I t G ) = 2t + n − 4 for t ≥ 1. As for the claim, we consider the following two cases.…”

“…Due to its difficulty, we have to focus on the cases when J G and I G are almost complete intersections. Luckily, graphs with almost complete intersection (parity) binomial edge ideals have been completely characterized by the nice work of Jayanthan et al in [18] and Kumar in [24]. Therefore, as the application of our main results, we can solve the linearization-of-regularity problems for those ideals when the underlying graph G is connected, with only one exception.…”

“…We will fix this isomorphism Φ throughout this section. And to compute the regularity of the powers of parity binomial edge ideals of some non-bipartite graphs, we need the following lemma from [24]. Lemma 5.5 ([24, Lemma 3.3]).…”

“…Thus, [17,Theorem 4.5] proved that reg(S/I G ) = n − 1. Notice that the natural generators of I G\e form a regular sequence, while after appending g e , one gets a d-sequence of length n by the proof of [24,Theorem 3.8]. Furthermore, (G \ e) e is obtained by attaching two paths to two vertices of a triangle respectively.…”

“…Actually, one can also consider the Lovász-Saks-Schrijver ideal L G ( [26]) and the permanental edge ideal Π G ( [13]) associated to G. However, by [24,Remark 3.4], we can focus only on binomial edge ideals and parity binomial edge ideals.…”