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Examining Kempe equivalence via commutative algebra

Hidefumi Ohsugi, Akiyoshi Tsuchiya

TL;DR

This work builds an algebraic framework to study Kempe equivalence of graph colorings by mapping colorings to binomial ideals arising from stable sets and 2-colorings. It proves that Kempe equivalence can be detected via membership in the 2-coloring ideal $J_G$ (equivalently in the quadratic part of the stable-set toric ideal), and extends this with the Kempe ideal $K_G$ to obtain complete representatives for $k$-Kempe classes using Gröbner-basis techniques. A key result is that the Hilbert function of the quotient by $K_G$ encodes the number of Kempe classes across induced subgraphs, enabling computation via algebraic invariants. The paper also provides a suite of algebraic algorithms for deciding Kempe equivalence, enumerating Kempe classes, and constructing explicit sequences of Kempe switches, thereby offering a uniform, computable framework for Kempe theory grounded in commutative algebra and toric geometry.

Abstract

Kempe equivalence is a classical and important notion on vertex coloring in graph theory. In the present paper, we introduce several ideals associated with graphs and provide a method to determine whether two $k$-colorings are Kempe equivalent via commutative algebra. Moreover, we give a way to compute all $k$-colorings of a graph up to Kempe equivalence by virtue of the algebraic technique on Gröbner bases. As a consequence, the number of $k$-Kempe classes can be computed by using Hilbert functions. Finally, we introduce several algebraic algorithms related to Kempe equivalence.

Examining Kempe equivalence via commutative algebra

TL;DR

This work builds an algebraic framework to study Kempe equivalence of graph colorings by mapping colorings to binomial ideals arising from stable sets and 2-colorings. It proves that Kempe equivalence can be detected via membership in the 2-coloring ideal (equivalently in the quadratic part of the stable-set toric ideal), and extends this with the Kempe ideal to obtain complete representatives for -Kempe classes using Gröbner-basis techniques. A key result is that the Hilbert function of the quotient by encodes the number of Kempe classes across induced subgraphs, enabling computation via algebraic invariants. The paper also provides a suite of algebraic algorithms for deciding Kempe equivalence, enumerating Kempe classes, and constructing explicit sequences of Kempe switches, thereby offering a uniform, computable framework for Kempe theory grounded in commutative algebra and toric geometry.

Abstract

Kempe equivalence is a classical and important notion on vertex coloring in graph theory. In the present paper, we introduce several ideals associated with graphs and provide a method to determine whether two -colorings are Kempe equivalent via commutative algebra. Moreover, we give a way to compute all -colorings of a graph up to Kempe equivalence by virtue of the algebraic technique on Gröbner bases. As a consequence, the number of -Kempe classes can be computed by using Hilbert functions. Finally, we introduce several algebraic algorithms related to Kempe equivalence.
Paper Structure (7 sections, 21 theorems, 60 equations, 7 figures, 6 algorithms)

This paper contains 7 sections, 21 theorems, 60 equations, 7 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Stable set ideals
  3. Gröbner bases
  4. 2-coloring ideals
  5. Examining Kempe equivalence
  6. Computing Kempe classes via Kempe ideals
  7. Algorithms

Key Result

Theorem 1.1

Let $G$ be a graph on $[d]$ and let $f,g$ be $k$-colorings of $G$. Then $f \sim_k g$ if and only if ${\mathbf x}_f-{\mathbf x}_g \in J_G$.

Figures (7)

  • Figure :
  • Figure : Computation of the reduced Gröbner basis of $K_G$
  • Figure : Determination of the Kempe equivalence
  • Figure : Computation of a complete representative system for ${\rm kc}(G,k)$
  • Figure : Enumeration of elements in a Kempe equivalent class
  • ...and 2 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: OTkempe
  • Proposition 2.2: OTkempe
  • Proposition 2.3
  • Example 3.1
  • Lemma 3.2: Buchberger's Criterion CLO
  • Lemma 3.3: CLO
  • Lemma 3.4: HHO
  • Lemma 3.5: HHO
  • ...and 28 more