Recognizing Graph Theoretic Properties with Polynomial Ideals

TL;DR

This paper explores how polynomial ideals and algebraic techniques can be used to detect complex graph properties like k-colorability, Hamiltonicity, and automorphism rigidity, advancing the polynomial method in combinatorics.

Contribution

It introduces new algebraic methods and algorithms for recognizing graph properties using polynomial ideals, expanding the toolkit for combinatorial problem solving.

Findings

Polynomial ideals can detect k-colorability and Hamiltonicity.

Algebraic certificates provide new ways to verify graph properties.

The methods involve diverse algebraic and geometric techniques.

Abstract

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Groebner bases, toric algebra, convex programming, and real algebraic geometry.