Kempe equivalence and quadratic toric rings
Hidefumi Ohsugi, Akiyoshi Tsuchiya
TL;DR
The paper studies the connection between Kempe equivalence in graph coloring and the quadraticity of stable set rings $K[G]$. It proves that $K[G]$ is quadratic if and only if, for every replication graph $H$ of any induced subgraph of $G$ and every $k \ge \chi(H)$, all $k$-colorings of $H$ are Kempe equivalent, establishing a bridge between coloring dynamics and toric algebra. This result ties into perfectly contractile graphs and suggests that Everett and Reed's conjecture on perfectly contractile graphs would imply the authors' conjecture that quadratic stable set rings correspond to Kempe-equivalent colorings on replication graphs; it also confirms quadraticity for several important graph classes, such as weakly chordal, Meyniel, and perfectly orderable graphs. Additionally, the authors propose a new purely combinatorial conjecture that characterizes perfectly contractile graphs via Kempe equivalence on replication graphs, linking combinatorial graph structure with algebraic properties of $K[G]$.
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.