Eliahou–Villarreal conjecture on projective dimension of edge ideals

Let $G$ be a finite simple graph with vertices $V(G)=\{x_1,\dots,x_n\}$, let $R=\mathbb{K}[x\mid x\in V(G)]$, and let $I(G)=(x_ix_j\mid \{x_i,x_j\}\in E(G))$ be its edge ideal. The quotient $R/I(G)=\mathbb{K}[\Delta_G]$ has a 2-linear resolution when its minimal graded free resolution has the form

$$0 \longrightarrow R(-2-p)^{\beta_p} \longrightarrow \cdots\longrightarrow R(-3)^{\beta_1} \longrightarrow R(-2)^{\beta_0} \longrightarrow R \longrightarrow \mathbb{K}[\Delta_G] \longrightarrow 0.$$

Eliahou–Villarreal conjecture. If the edge ideal of a connected graph $G$ has a 2-linear resolution, then

$$\operatorname{pdim}(R/I(G))=\max_{1\leq i\leq n}\{\deg_G(x_i)\}.$$

By Fröberg's theorem, the 2-linear-resolution hypothesis is equivalent to $G$ being co-chordal in the convention used here. The paper states that this conjecture is false in general and constructs counterexamples; it also proves the equality for several subclasses, including sequentially Cohen–Macaulay co-chordal graphs and graphs with a full vertex.