On the Linearity of Squarefree Powers of Edge Ideals

Abstract: Let $G$ be a graph and $I(G)$ its edge ideal. The $p$-th squarefree power $I(G)^{[p]}$ is the monomial ideal generated by squarefree monomials corresponding to the matchings of size $p$ of $G$. In this paper, we provide a combinatorial characterization of when $I(G)^{[p]}$ is linearly related, i.e., when its first syzygy module is generated by linear forms. Moreover, for a $1$-dimensional flag simplicial complex $Δ$ and its Stanley-Reisner ideal $I_Δ$, which arises as the edge ideal of the complement graph of $Δ$, we describe the shape of the Betti table of $I_Δ^{[p]}$ and we give a combinatorial characterization of when $I_Δ^{[p]}$ has a linear resolution.

• The paper provides a combinatorial characterization for when squarefree powers are linearly related based on forbidden induced subgraph criteria.
• It outlines explicit formulas for extremal Betti numbers using counts of specific subgraphs like complete bipartite and crown graphs.
• The study shows how graph-theoretic properties of edge ideals determine homological invariants, offering practical conditions for linear resolutions.

Linearity of Squarefree Powers of Edge Ideals

Overview

The paper "On the Linearity of Squarefree Powers of Edge Ideals" (2607.00838) addresses the homological and combinatorial properties of squarefree powers of edge ideals. The main focus is on characterizing the linearity, linear relations, and Betti table shape of these ideals, with significant attention to the connections between the structure of the underlying graphs (or, equivalently, 1-dimensional flag simplicial complexes) and the algebraic properties of their associated squarefree powers.

The work provides concrete combinatorial criteria for when squarefree powers are linearly related or have linear resolutions, identifies precise obstructions in terms of forbidden induced subgraphs (notably complete bipartite graphs and crown graphs), and gives explicit formulas for certain extremal Betti numbers in terms of subgraph count. The implications extend to the understanding of homological invariants of squarefree monomial ideals and offer new insights into the interplay between combinatorial graph theory and the minimal graded free resolutions of monomial ideals.

Squarefree Powers of Edge Ideals: Notation and Motivation

Let $G$ be a simple graph with edge ideal $I(G) \subset S = K[x_1, ..., x_n]$, i.e., $I(G)$ is generated by the squarefree quadratic monomials corresponding to edges of $G$. The $p$-th squarefree power $I(G)^{[p]}$ consists of all squarefree monomials among the generators of $I(G)^p$. Equivalently, the generators of $I(G)^{[p]}$ correspond to $p$-matchings in $G$ (collections of $I(G) \subset S = K[x_1, ..., x_n]$0 disjoint edges). This construction ties algebraic and combinatorial properties: the resolution of $I(G) \subset S = K[x_1, ..., x_n]$1 reflects the complexity of the matching structure of $I(G) \subset S = K[x_1, ..., x_n]$2.

For a 1-dimensional flag simplicial complex $I(G) \subset S = K[x_1, ..., x_n]$3 (a triangle-free graph), the Stanley-Reisner ideal $I(G) \subset S = K[x_1, ..., x_n]$4 is the edge ideal of the complement of $I(G) \subset S = K[x_1, ..., x_n]$5. The study unites classical combinatorial commutative algebra (e.g., works of Villarreal and Fröberg) with advanced analysis of homological invariants for powers and squarefree powers of edge ideals, such as Castelnuovo-Mumford regularity, projective dimension, and the precise structure of their minimal free resolutions.

Main Results and Characterizations

Linearly Related Squarefree Powers

A central contribution is the combinatorial characterization for when $I(G) \subset S = K[x_1, ..., x_n]$6 is linearly related, i.e., its first syzygies are generated in degree $I(G) \subset S = K[x_1, ..., x_n]$7. The main theorem states:

$I(G) \subset S = K[x_1, ..., x_n]$8 is linearly related if and only if $I(G) \subset S = K[x_1, ..., x_n]$9 contains no induced subgraph $I(G)$0 on $I(G)$1 vertices with $I(G)$2 disconnected and matching number $I(G)$3.

For 1-dimensional flag complexes (triangle-free graphs) $I(G)$4, this translates into a forbidden induced subgraph criterion involving complete bipartite graphs:

• $I(G)$5 is linearly related iff $I(G)$6 has no induced subgraph isomorphic to $I(G)$7 for any even $I(G)$8 with $I(G)$9.

This characterization is proven using both the structure of the lcm-lattice of squarefree monomial ideals and detailed analysis of the associated graph of minimal generators, generalizing and correcting previous conjectures from the literature.

Linear Resolution of Squarefree Powers

A significantly more stringent property is the existence of a linear resolution (all syzygies are as linear as possible). The authors provide an exact combinatorial criterion:

$G$0 has a linear resolution if and only if $G$1 contains neither an induced subgraph isomorphic to any $G$2 ($G$3 even, $G$4) nor an induced subgraph isomorphic to the crown graph $G$5.

Here, the appearance of a crown graph serves as a new type of obstruction, only affecting higher syzygy degrees beyond the first. The proofs exploit Hochster's formula and combinatorial topology methods, relating the existence of certain homology cycles in simplicial complexes associated to the ideals to the existence of forbidden subgraphs.

Betti Numbers and Resolution Shape

The Betti table for $G$6 is analyzed in depth. Notably, the paper provides:

• Explicit enumerative formulas for extremal Betti numbers in terms of the counts of forbidden subgraphs. For instance,

$G$7

counts the obstructions to linear first syzygies.

• A similar formula holds for the Betti number

$G$8

when there are no $G$9 subgraphs.

• The full possible shapes of the graded Betti table are classified depending on the presence of the specified forbidden induced subgraphs.

Additionally, the regularity and projective dimension of $p$0 are shown to take only two possible values ($p$1 or $p$2 for regularity), closely tied to the matching structure or appearance of particular bipartite subgraphs.

Implications and Extensions

These characterizations have both algebraic and combinatorial implications:

Theoretical Implications

• Sharp transition points are identified for when the squarefree powers gain or lose desirable homological properties, governed by explicit graph-theoretic conditions.
• The classification demonstrates that, for large enough $p$3, the squarefree powers often become extremely well-behaved, recovering results of prior works for the last nontrivial squarefree power (e.g., Bigdeli-Herzog-Zaare-Nahandi).
• The interplay between the forbidden induced subgraph types ($p$4 for failure of first syzygy linearity; $p$5 for deeper syzygy nonlinearities) clarifies and systematizes the numerous empirical observations and special cases in previous literature.

Practical Implications

• These results provide effective, combinatorially checkable criteria for algebraists to use in determining linearity or minimal resolution shape of squarefree powers, aiding in computation and classification of monomial ideals arising from combinatorial objects.
• In applications where the Stanley-Reisner ring structure governs topological or combinatorial invariants (e.g., in combinatorics, algebraic statistics, or algebraic combinatorics), these formulas allow for the rapid assessment of complexity and algebraic regularity.

Future Developments

• The paper closes with open questions for the case when the complement of $p$6 contains triangles (i.e., for general graphs rather than just triangle-free graphs), where the presence of triangles complicates the forbidden subgraph characterization.
• Preliminary computational evidence suggests possible monotonicity for higher squarefree powers, i.e., that linearity at some level propagates upward, a phenomenon yet to be fully formalized in the general case.

Conclusion

This paper achieves a full combinatorial characterization of when squarefree powers of edge ideals and associated Stanley-Reisner ideals of 1-dimensional flag complexes are linearly related or have linear resolutions, in terms of explicit forbidden subgraphs. It further provides precise enumerative correspondences for certain Betti numbers and a detailed classification of the minimal resolution shape, blending topological, algebraic, and combinatorial methods. The work lays a solid groundwork for future exploration of squarefree powers and their homological invariants beyond the triangle-free case, posing several challenging questions for further research.