Linear quotients of connected ideals of graphs

TL;DR

This work extends the theory of edge ideals to $t$-connected ideals $J_t(G)$, generated by monomials corresponding to $t$-connected vertex sets. It establishes a sharp combinatorial–algebraic correspondence for chordal graphs: $J_t(G)$ has a linear resolution if and only if $G$ is $t$-gap-free, equivalently $J_t(G)$ has linear quotients. It further shows that for gap-free and $t$-claw-free graphs (with $t\ge3$), $J_t(G)$ has linear quotients and hence linear resolution, while non-chordal examples (cycles) illustrate the necessity of structural assumptions. The paper also connects $J_t(G)$ to the Stanley-Reisner viewpoint via the independence complex, discusses equality with the $t$-path ideal in claw-free graphs for small $t$, and raises open questions about broader classifications and vertex-splittable properties.

Abstract

As a higher analogue of the edge ideal of a graph, we study the $t$-connected ideal $\operatorname{J}_{t}$. This is the monomial ideal generated by the connected subsets of size $t$. For chordal graphs, we show that $\operatorname{J}_{t}$ has a linear resolution iff the tree is $t$-gap-free, and that this is equivalent to having linear quotients. We then show that if $G$ is any gap-free and $t$-claw-free graph, then $\operatorname{J}_{t}(G)$ has linear quotients and, hence, linear resolution.

Theorems & Definitions (45)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Proposition 3.1