Cohen-Macaulayness of squarefree powers of edge ideals of whisker graphs

Abstract

Let $G$ be a finite simple graph with edge ideal $I(G)$. For $q\ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ is generated by products of $q$ pairwise disjoint edges of $G$. It is the Stanley-Reisner ideal of a simplicial complex $\mathsf{MF}^q(G)$, called the $q$-matching-free complex, whose faces are those subsets $F\subseteq V(G)$ for which the induced subgraph $G[F]$ contains no matching of size $q$. We study $\mathsf{MF}^q(G)$ when $G=W(H)$ is a whisker graph. We first characterize purity. If $H$ is bipartite, then $\mathsf{MF}^q(G)$ is pure for all $q$. Otherwise, let $\ell$ denote the length of the smallest odd cycle of $H$ and set $n=|V(H)|$. Then $\mathsf{MF}^q(G)$ is pure if and only if $q<\lceil \ell/2\rceil$ or $q>n-\lfloor \ell/2\rfloor.$ We next determine the exact range of shellability. Let $m=\operatorname{girth}(H)$, with $m=\infty$ if $H$ is acyclic. Then $\mathsf{MF}^q(G)$ is shellable for \[ 1\le q\le \begin{cases} \lceil m/2\rceil, & \text{if } m<\infty,\\ \nu(G), & \text{if } m=\infty. \end{cases} \] Consequently, $I(G)^{[q]}$ is Cohen-Macaulay for $1\le q\le\lfloor m/2\rfloor$ when $m<\infty$, and for all $1\le q\le\nu(G)$ when $m=\infty$. If $m$ is odd, then $I(G)^{[q]}$ is sequentially Cohen-Macaulay for $q=\lceil m/2\rceil$. We further obtain extremal characterizations: $\mathsf{MF}^{2}(G)$ is Cohen-Macaulay if and only if $H$ has no induced $3$-cycle, and $\mathsf{MF}^{\,n-1}(G)$ is Cohen-Macaulay if and only if $H$ is acyclic. Finally, we compute the depth of $I(G)^{[q]}$ for whisker graphs and verify a conjecture on the depth of squarefree powers of whisker cycles in the relevant range.