Connected ideals of chordal graphs

Abstract: For $t\geq 2$, the $t$-independence complex 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 Stanley-Reisner ideal $I_{t}(G)$ of the $t$-independence complex of $G$, called $t$-connected ideal, is generated by monomials in a polynomial ring $R$ corresponding to all $A\subseteq V(G)$ of size $t$ such that $G[A]$ is connected. This class of ideals is a natural generalization of the edge ideals of graphs. In this paper, we investigate the $t$-connected ideals of chordal graphs. In particular, we prove that for a chordal graph $G$ and for all $t$ [ \mathrm{reg}(R/I_{t}(G))=(t-1)\nu_{t}(G) \text{ and } \mathrm{pd}(R/I_{t}(G))=\mathrm{bight}(I_{t}(G)), ] where $\nu_{t}(G)$ denotes the induced matching number of the corresponding hypergraph of $I_{t}(G)$, 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 completely characterize when the $t$-connected ideal 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.