On the Castelnuovo-Mumford regularity of squarefree powers of edge ideals

Abstract

Assume that G is a graph with edge ideal I(G) and matching number match(G). For every integer s ≥ 1, we denote the s-th squarefree power of I(G) by I(G) [s] . It is shown that for every positive integer s ≤ match(G), the inequality reg(I(G) [s] ) ≤ match(G) + s holds provided that G belongs to either of the following classes: (i) very well-covered graphs, (ii) semi-Hamiltonian graphs, or (iii) sequentially Cohen-Macaulay graphs. Moreover, we prove that for every Cameron-Walker graph G and for every positive integer s ≤ match(G), we have reg(I(G) [s] ) = match(G) + s