Regularity of powers of bipartite graphs

Abstract

Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all s ≥ 1, we obtain upper bounds for reg(I(G) s ) for bipartite graphs. We then compare the properties of G and G ′ , where G ′ is the graph associated with the polarization of the ideal (I(G) s+1 : e 1 · · · e s ), where e 1 , . . . e s are edges of G. Using these results, we explicitly compute reg(I(G) s ) for several subclasses of bipartite graphs.

Regularity of Edge Ideals and Their Powers

“…Fortunately, it turns out that (I(G) s+1 : m) is generated by monomials in degree 2. After polarization, the ideal (I(G) s+1 : m) can be viewed as the edge ideal of a graph that is obtained from G by adding even-connected edges with respect to m. Some of the combinatorial properties of this ideal in relation to G are studied in [37].…”

“…A similar general upper bound generalizing that of Theorem 4.6 is, unfortunately, not available. It is established, by Jayanthan, Narayanan and Selvaraja [37], only for a special class of graphs -bipartite graphs. Sketch of proof.…”

Improved bounds for the regularity of edge ideals of graphs

“…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].…”

Regularity and projective dimension of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs

“…If w j = 1 for all j, then I(D) recovers the usual edge ideal of its (undirected) underlying graph. Edge ideals of (undirected) graphs have been investigated extensively in the literature [1,2,3,4,5,6,7,14,19,21,24,25,29]. In general, edge ideals of weighted oriented graphs are different from edge ideals of edge-weighted (undirected) graphs defined by Paulsen and Sather-Wagstaff [27].…”

“…In this regard, there has been an interest in determining the smallest value t 0 such that pd (S/I t ) is a constant for all t ≥ t 0 . (see [14,19,21,25]).…”