Square-free powers of Cohen-Macaulay forests, cycles, and whiskered cycles

Let $I(G{)}^{[k]}$ denote the $k^{th}$ square-free power of the edge ideal $I(G)$ of a graph $G$. In this article, we provide a precise formula for the depth of $I(G{)}^{[k]}$ when $G$ is a Cohen-Macaulay forest. Using this, we show that for a Cohen-Macaulay forest $G$, the $k^{th}$ square-free power of $I(G)$ is always Cohen-Macaulay, which is quite surprising since all ordinary powers of $I(G)$ can never be Cohen-Macaulay unless $G$ is a disjoint union of edges. Next, we give an exact formula for the regularity and tight bounds on the depth of square-free powers of edge ideals of cycles. In the case of whiskered cycles, we obtain tight bounds on the regularity and depth of square-free powers, which aids in identifying when such ideals have linear resolutions. Additionally, we compute depth of $I(G{)}^{[2]}$ when $G$ is a cycle or whiskered cycle, and regularity of $I(G{)}^{[2]}$ when $G$ is a whiskered cycle.