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 $\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}(G)$. For every integer $s\ge 1$, we denote the $s$-th squarefree power of $I(G)$ by $I(G{)}^{[s]}$. It is shown that for every positive integer $s\le \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}(G)$, the inequality $\mathrm{r}\mathrm{e}\mathrm{g}(I(G{)}^{[s]})\le \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}(G)+s$ 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 $s\le \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}(G)$, we have $\mathrm{r}\mathrm{e}\mathrm{g}(I(G{)}^{[s]})=\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}(G)+s$