On the Linearity of Squarefree Powers of Edge Ideals

Let $G$ be a graph and $I(G)$ its edge ideal. The $p$-th squarefree power $I(G{)}^{[p]}$ is the monomial ideal generated by squarefree monomials corresponding to the matchings of size $p$ of $G$. In this paper, we provide a combinatorial characterization of when $I(G{)}^{[p]}$ is linearly related, i.e., when its first syzygy module is generated by linear forms. Moreover, for a $1$-dimensional flag simplicial complex $\mathrm{\Delta }$ and its Stanley-Reisner ideal $I_{\mathrm{\Delta }}$, which arises as the edge ideal of the complement graph of $\mathrm{\Delta }$, we describe the shape of the Betti table of $I_{\mathrm{\Delta }}^{[p]}$ and we give a combinatorial characterization of when $I_{\mathrm{\Delta }}^{[p]}$ has a linear resolution.