Kempe equivalence and quadratic toric rings

Abstract

Kempe equivalence is a classical and fundamental notion in graph coloring theory. In the present paper we establish a connection between Kempe equivalence and quadratic stable set ring, which are toric rings associated to graphs. In fact, we characterize when the stable set ring of a graph is quadratic by using Kempe equivalence. As an application, we relate our theorem to the theory of perfectly contractile graphs, a hereditary subclass of perfect graphs introduced by Bertschi. In particular, our characterization implies that the conjecture of Everett and Reed on perfectly contractile graphs entails the conjecture of the authors and Shibata on quadratic stable set rings. Furthermore, we show that the stable set rings of several important subclasses of perfectly contractile graphs including weakly chordal graphs are quadratic. Finally, we propose a new combinatorial conjecture characterizing perfectly contractile graphs purely in terms of Kempe equivalence on replication graphs.