Linear resolution of connected graph ideals and their powers

TL;DR

This paper studies the linear resolutions of connected graph ideals and their powers, introducing new graph classes where these ideals have desirable algebraic properties, extending classical results in combinatorial commutative algebra.

Contribution

It introduces the class of co-chordal-cactus graphs and proves linear resolutions for their connected ideals and powers, extending known results to broader graph families.

Findings

Connected ideals have linear resolutions in co-chordal-cactus graphs.

Powers of connected ideals also have linear resolutions in several graph families.

Edge ideals have bounded Castelnuovo-Mumford regularity in specific graph classes.

Abstract

For a finite simple graph $G$ and an integer $r \ge 1$, the $r$-connected ideal $I_r(G)$ is the squarefree monomial ideal generated by the vertex sets of connected induced subgraphs of size $r+1$, extending the classical edge ideal. We investigate the linearity of the minimal free resolutions of $I_r(G)$ via structural features of the associated clutter $\mathcal{C}_r(G)$. We introduce the class of co-chordal-cactus graphs and prove that $I_r(G)$ has a linear resolution for all $r \ge 2$ whenever $G$ lies in this family. The result further extends to $(2K_2, C_4)$-free graphs and co-grid graphs. For $r=1$, we show that the edge ideal $I_1(G)$ has Castelnuovo-Mumford regularity at most $3$ for all co-chordal-cactus and co-grid graphs. We also examine powers of connected ideals and establish that $I_r(G)^q$ has a linear resolution for every $q \ge 1$ in several natural graph families, including complements of trees with bounded degree, complete multipartite graphs, complements of cycles, graphs obtained by gluing complete graphs along cliques, and certain subclasses of split graphs.