Weak Lefschetz property for whiskered graphs with independence number at least three

Prove that for every graph G with independence number at least 3, the artinian algebra obtained from the edge ideal of the whiskered graph w(G) by adjoining the squares of all vertex variables does not have the weak Lefschetz property.

Background

The paper studies artinian algebras associated with whiskered graphs, whose edge ideals define level algebras after imposing square relations on the variables. The weak Lefschetz property concerns maximal rank of multiplication by a general linear form between consecutive graded components; for these monomial algebras, it suffices to test the sum of the variables.

The authors prove the conjecture for bipartite graphs and establish failure of the weak Lefschetz property under the stronger numerical condition that the independence number is at least n/3+2 for a graph on n vertices. They also report computational verification for all graphs on at most seven vertices and many graphs on eight vertices, but the general case of independence number at least three remains unresolved.

References

Let $G$ be a graph with independence number at least $3$ and $I$ the edge ideal of its whiskered graph $w(G)$. Then the algebra $$ \frac{K[x_1, \dots, x_{2n}]}{I + (x_12, \dots, x_{2n}2)} does not have the weak Lefschetz property.

— Roller Coaster Gorenstein algebras and Koszul algebras failing the weak Lefschetz property  (2502.00155 - Holleben et al., 31 Jan 2025) in Conjecture 1.5, Section 1 (Introduction); also stated as Conjecture 3.8 in Section 3