Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles

Abstract

Let $G$ be a finite simple graph and let $I(G)$ denote its edge ideal. For $q\ge 1$, the $q$-th squarefree power $I(G{)}^{[q]}$ is generated by squarefree monomials corresponding to matchings of size $q$ in $G$. We denote by $\mathrm{reg}(-)$ the Castelnuovo-Mumford regularity. Das, Roy, and Saha conjectured that if $G=W(C_n)$ is a whiskered cycle, then \[ \operatorname{reg}\big(I(G)^{[q]}\big) = 2q + \left\lfloor \frac{n - q - 1}{2} \right\rfloor ~ \text{for all } 1 \le q \le ν(G), \] where $\nu (G)$ denotes the matching number of $G$. In this paper, we confirm this conjecture by determining the exact value of $\mathrm{reg}(I(G{)}^{[q]})$.

14 Pages. Comments are welcome