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

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.