Regularity of path ideals of gap free graphs

Abstract

Abstract. In this paper we study the Castelnuovo-Mumford regularity of the path ideals of finite simple graphs. We find new upper bounds for various path ideals of gap free graphs. In particular we prove that the path ideals of gap free and claw graphs have linear minimal free resolutions.

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

“…In this section, we gather together needed definitions and basic facts, which will be used throughout this paper. However, for more details, we refer the reader to [4,10,15,17,20,26,28,32,34,35].…”

“…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 finding the exact form of the linear function and determining the stabilization index t 0 where reg(I t ) becomes linear (cf. [1,2,3,4,5,6,7,21,24,29]). It turns out that even in the case of monomial ideals it is challenging to find the linear function and t 0 (cf.…”

“…The weight function ofD is: w(x 1 ) = • • • = w(x 4 ) = 1 and w(y 1 ) = • • • = w(y 4 ) = 2. Then the edge ideal of D is I(D) = (x 1 y 2 1 , x 1 y 2 2 , x 2 y 2 1 , x 2 y 2 2 , x 3 y 2 3 , x 4 y 2 3 , x 4 y24 ). By using CoCoA, we obtain pd (I(D) 2 ) = 5 = 8 − 2 − 1.…”