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

TL;DR

This paper investigates the Cohen-Macaulayness and shellability of squarefree powers of edge ideals of whisker graphs, providing characterizations based on graph properties like bipartiteness, cycles, and girth.

Contribution

It offers new characterizations of purity, shellability, and Cohen-Macaulayness for squarefree powers of edge ideals in whisker graphs, linking algebraic properties to graph structures.

Findings

$ ext{MF}^q(G)$ is pure if $H$ is bipartite or under certain cycle length conditions.

$ ext{MF}^q(G)$ is shellable within a specific range depending on girth or independence number.

$I(G)^{[q]}$ is Cohen-Macaulay or sequentially Cohen-Macaulay under conditions related to cycles and girth.

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.