Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles

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 $$\mathrm{reg}(I(G{)}^{[q]})=2q+\lfloor \frac{n-q-1}{2}\rfloor \text{ for all }1\le q\le \nu (G),$$ where $\nu (G)$ denotes the matching number of $G$. In this paper, we confirm this conjecture by determining the exact value of $\mathrm{reg}(I(G{)}^{[q]})$.