Edge ideals with linear quotients and without homological linear quotients

A monomial ideal $I$ is said to have homological linear quotients if for each $k\geq 0$, the homological shift ideal $\mathrm{HS}_k(I)$ has linear quotients. It is a well-known fact that if an edge ideal $I(G)$ has homological linear quotients, then $G$ is co-chordal. We construct a family of co-chordal graphs ${\mathrm{H}n^c}{n\geq 6}$ and propose a conjecture that an edge ideal $I(G)$ has homological linear quotients if and only if $G$ is co-chordal and $\mathrm{H}_n^c$-free for any $n\geq 6$. In this paper, we prove one direction of the conjecture. Moreover, we study possible patterns of pairs $(G,k)$ of a co-chordal graph $G$ and integer $k$ such that $\mathrm{HS}_k(I(G))$ has linear quotients.

💡 Research Summary

The paper investigates a refined homological property of edge ideals of graphs, namely “homological linear quotients.” For a monomial ideal I, the k‑th homological shift ideal HSₖ(I) is generated by the multidegrees appearing in the k‑th step of a minimal free resolution of I. An ideal I is said to have homological linear quotients if every HSₖ(I) (for all k ≥ 0) possesses linear quotients, a combinatorial condition that guarantees a linear resolution. It is classical (Fröberg) that if an edge ideal I(G) has a linear resolution then the complement graph Gᶜ is chordal; equivalently, G itself is co‑chordal. However, co‑chordality alone does not ensure that all homological shift ideals of I(G) have linear quotients, as shown by a counterexample in earlier work.

To capture the missing obstruction, the authors introduce a family of graphs {Hₙ}ₙ≥6. Each Hₙ is built from a long path together with a few extra edges that create a “star‑like” attachment at the end of the path. The complement Hₙᶜ is then a co‑chordal graph whose homological shift ideals fail to have linear quotients already for k = 2. The authors denote by Hₙᶜ‑free the class of graphs that do not contain any Hₙᶜ as an induced subgraph.

Conjecture 1.1 (the central conjecture of the paper) states that an edge ideal I(G) has homological linear quotients if and only if G is co‑chordal and Hₙᶜ‑free for all n ≥ 6. The paper proves the “only‑if” direction (Theorem 5.1): if I(G) has homological linear quotients then G must be co‑chordal and cannot contain any Hₙᶜ. The converse is proved only for graphs with at most ten vertices; the general case remains open.

Beyond the conjecture, the authors explore the possible patterns of k for which HSₖ(I(G)) fails or succeeds to have linear quotients when G is a fixed co‑chordal graph. It is known that HS₀(G) and HS₁(G) always have linear quotients. The paper shows that for any integers 2 ≤ s ≤ t, one can construct a co‑chordal graph G exhibiting three distinct behaviours:

1. Pattern A (Theorem 3.1 / Theorem 1.3): HSₖ(G) lacks linear quotients for 2 ≤ k < s, has linear quotients for s ≤ k ≤ t, and becomes the zero ideal for k > t.

2. Pattern B (Theorem 4.1 / Theorem 1.4): HSₖ(G) has linear quotients for 2 ≤ k < s, fails for s ≤ k ≤ t, and vanishes for k > t.

3. Pattern C (Theorem 4.2 / Theorem 1.5): HSₖ(G) has linear quotients for all k except a single “bad” value k = s, where it fails; again the ideal disappears after t.

These constructions rely on two auxiliary families of graphs. For each n ≥ 6 and r ≥ 0, the authors define LHₙ,ᵣ as Hₙ together with r new “leaf” vertices attached to a distinguished vertex (the vertex labelled n‑3). The complement LHₙ,ᵣᶜ is still chordal, and the perfect elimination ordering x₁ > x₂ > … > xₙ₊ᵣ holds. Using Fröberg’s description of homological shift ideals for chordal graphs (Lemma 2.2), the authors explicitly list the minimal generators of HSₖ(LHₙ,ᵣᶜ) in terms of six types (I–VI) based on the positions of the smallest and largest variables in the monomial. Proposition 3.4 shows that for Hₙᶜ the generators are precisely of types I–V.

A key technical tool is Lemma 2.5, which says that if a chordal graph G has HSₖ(I(Gᶜ)) with linear quotients, then any induced subgraph H inherits this property. This allows the authors to reduce non‑linearity arguments to smaller forbidden subgraphs. Lemma 2.6 provides a concrete non‑linear example: the ideal generated by all monomials xᵢxₐx_b (i ranging) together with all monomials xᵢx_cx_d fails to have a linear resolution, and consequently fails to have linear quotients. By embedding such a configuration into HSₖ(LHₙ,ᵣᶜ) for appropriate k, the authors prove the failure of linear quotients in the desired ranges.

Lemma 2.7 gives a lexicographic criterion for linear quotients: an ideal generated in a single degree has linear quotients with respect to the lex order if and only if for any two generators f <ₗₑₓ g there exist variables x_i∈supp(f) and x_j∈supp(g)\supp(f) with j < i such that x_j·f/x_i is again a minimal generator. The authors verify this condition for the “good” ranges of k, and show its violation for the “bad” ranges, thereby establishing the three patterns.

The paper concludes with a discussion of open problems. The main conjecture’s converse for arbitrary co‑chordal graphs remains unresolved. Understanding the relationship between the Betti numbers of HSₖ(I(G)) and the combinatorial structure of G (e.g., treewidth, clique number) is highlighted as a promising direction. Extending the theory to hypergraphs or to monomial ideals arising from higher‑dimensional simplicial complexes is also suggested.

Overall, the work makes a substantial contribution by identifying a precise forbidden induced subgraph (Hₙᶜ) that blocks homological linear quotients, proving one direction of the resulting characterization, and demonstrating that the behavior of homological shift ideals can be arbitrarily intricate even within the class of co‑chordal graphs. This opens new avenues for exploring the interplay between combinatorial graph properties and higher‑order homological invariants of monomial ideals.