Stanley-Reisner ideals of higher independence complexes of chordal graphs

Abstract

For $t\geq 2$, the $t$-independence complex $\mathrm{Ind}_t(G)$ of a graph $G$ is the collection of all $A\subseteq V(G)$ such that each connected component of the induced subgraph $G[A]$ has at most $t-1$ vertices. The topology of $\mathrm{Ind}_t(G)$ is intimately related to the combinatorial property of $G$. In this article, we consider the Stanley-Reisner ideal $J_{t}(G)$ of $\mathrm{Ind}_t(G)$ and focus on its algebraic properties. We prove that for a chordal graph $G$ and for all $t$ \[ \mathrm{reg}(R/J_{t}(G))=(t-1)\nu_{t}(G) \text{ and } \mathrm{pd}(R/J_{t}(G))=\mathrm{bight}(J_{t}(G)), \] where $\nu_{t}(G)$ denotes the induced matching number of the corresponding hypergraph of $J_{t}(G)$, and $\mathrm{reg}$, $\mathrm{pd}$ and $\mathrm{bight}$ stand for the regularity, projective dimension, and big height, respectively. As a consequence of the above results, we combinatorially characterize when the Stanley-Reisner ideal of the $t$-independence complex of a chordal graph has a linear resolution as well as when it satisfies the Cohen-Macaulay property. The above formulas and their consequences can be seen as a nice generalization of the classical results corresponding to the edge ideals of chordal graphs.