Squarefree Powers of Edge Ideals under Graph Joins and Cones

Abstract

For $q\ge 1$, the $q$-th squarefree power $I(G{)}^{[q]}$ of the edge ideal of a graph $G$ is generated by the squarefree monomials supported on $q$-matchings of $G$; it is the Stanley--Reisner ideal of the complex ${\Delta }_q(G)=\{F\subseteq V(G):\nu (G[F])<q\}$, where $\nu$ denotes matching number. We prove a general formula for the matching number of an arbitrary graph join, \[ ν(G\ast H) = \min\Big(ν(G)+|V(H)|,\ \ ν(H)+|V(G)|,\ \ \Big\lfloor\tfrac{|V(G)|+|V(H)|}{2}\Big\rfloor\Big), \] via the Tutte--Berge formula, and use it to decompose ${\Delta }_q(G\ast H)$ for arbitrary graphs $G,H$. Specializing to the wheel graph ${\mathcal{W}}_n={\mathcal{C}}_n\ast {\mathcal{K}}_1$, we determine the Krull dimension and height of $R/I({\mathcal{W}}_n{)}^{[q]}$ exactly for all $n\ge 3$, $1\le q\le \lfloor n/2\rfloor$, and -- combining our matching-number computations with a recent Tutte-type Cohen-Macaulayness criterion of Ficarra and Moradi -- prove that at the \emph{top} squarefree power $q=\nu ({\mathcal{W}}_n)=\lceil n/2\rceil$, the ideal $I({\mathcal{W}}_n{)}^{[\nu ({\mathcal{W}}_n)]}$ is literally the squarefree Veronese ideal, so that $R/I({\mathcal{W}}_n{)}^{[\nu ({\mathcal{W}}_n)]}$ is Cohen-Macaulay with \[ {\rm dim} = {\rm depth} = {\rm reg}\big(R/I(\mathcal{W}_n)^{[ν(\mathcal{W}_n)]}\big) = 2\Big\lceil\frac n2\Big\rceil-1. \] This resolves all four classical invariants at the top power, and confirms there the pattern depth$(R/I({\mathcal{W}}_n{)}^{[q]})=2q-1$ that our computational data (now extended to $n\le 13$, every valid $q$) suggests holds throughout. We prove a general depth formula for squarefree powers of cone graphs, via a Betti-splitting exact sequence, that reduces this pattern to two more tractable statements about the underlying cycle alone; both are verified computationally in every case checked but left open in general.