Perfectly contractile graphs and quadratic toric rings

TL;DR

This paper explores a special class of perfect graphs characterized by forbidden induced subgraphs and connects their combinatorial properties to the algebraic structure of associated toric ideals, proposing a conjecture linking graph classes to quadratic generation.

Contribution

It introduces a conjecture linking perfect graphs in class ${ mf A}$ to quadratic generation of their toric ideals and verifies this for several subclasses of perfectly contractile graphs.

Findings

Conjecture holds for Meyniel graphs.

Conjecture holds for perfectly orderable graphs.

Conjecture holds for clique separable graphs.

Abstract

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class ${\mathcal A}$ of graphs that have no odd holes, no antiholes and no odd stretchers as induced subgraphs. In particular, every graph belonging to ${\mathcal A}$ is perfect. Everett and Reed conjectured that a graph belongs to ${\mathcal A}$ if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to ${\mathcal A}$ from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph $G$ belongs to ${\mathcal A}$ if and only if the toric ideal of the stable set polytope of $G$ is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.