An algebraic reduction of Hedetniemi's conjecture

TL;DR

This paper reduces Hedetniemi's conjecture to an algebraic problem involving polynomial ideals and demonstrates computational experiments using Gröbner bases to explore partial solutions.

Contribution

It introduces an algebraic reduction of Hedetniemi's conjecture to ideal inclusion problems and applies computational algebra techniques to investigate potential solutions.

Findings

Reduction of Hedetniemi's conjecture to polynomial ideal inclusion

Implementation of Gröbner basis computations for partial solutions

Initial computational experiments suggest new avenues for research

Abstract

For a graph $G$, let $\chi (G)$ denote the chromatic number. In graph theory, the following famous conjecture posed by Hedetniemi has been studied: For two graphs $G$ and $H$, $\chi (G\times H)=\min\{\chi (G),\chi (H)\}$, where $G \times H$ is the tensor product of $G$ and $H$. In this paper, we give a reduction of Hedetniemi's conjecture to an inclusion relation problem on ideals of polynomial rings, and we demonstrate computational experiments for partial solutions of Hedetniemi's conjecture along such a strategy using Gr\"{o}bner basis.