Regularity of Edge Ideals and Their Powers

Abstract

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of reg I(G) and the asymptotic linear function reg I(G) q , for q ≥ 1, in terms of combinatorial data of the given graph G.

Regularity of symbolic powers of edge ideals of unicyclic graphs

“…Also, it is not difficult to check that I(G) (3) ⊆ I(G) 2 . Therefore, Corollary 3.9 easily follows from Theorem 3.6 and the above mentioned result of Banerjee and Nevo [6]. Indeed, we prove something more.…”

“…This implies that z x u = z x xyx ′ y ′ ∈ p 4 C , for every C ∈ C(G). Thus, z x u ∈ I(G) (4) . Consequently, z x ∈ X 0 , and therefore, we have u ∈ (X 0 ).…”

“…If C ′ ∩ (A ∪ {x, y}) = {x}, then it follows from vx 2 y 2 ∈ I(G) (4) ⊆ p 4 C ′ that 2 = deg x (vx 2 y 2 ) ≥ 4, which is a contradiction. Therefore,…”

“…As u is in the set of minimal monomial generators of I(G) 2 , we may write u = e 1 e 2 , for some edges e 1 , e 2 ∈ E(G). Let v be a monomial in (I(G) (4) : u). If v ∈ I(G), then v belongs to (I(G) 3 : u) and we are done.…”

“…Recently, Banerjee and Nevo [6] proved that for every graph G, we have reg(I(G) 2 ) ≤ reg(I(G)) + 2. Thus, confirming Conjecture 1.1 in the case of s = 2.…”