A survey on Hedetniemi's conjecture

TL;DR

This survey reviews the history, recent disproof, and ongoing open problems related to Hedetniemi's conjecture in graph theory, highlighting key developments and methodologies used in counterexample constructions.

Contribution

The paper provides a comprehensive overview of Hedetniemi's conjecture, summarizing the proof of its disproof and discussing the techniques behind counterexample discovery.

Findings

Hedetniemi's conjecture is false for all n ≥ 4.

The conjecture holds for n ≤ 3.

Counterexamples have been constructed using novel combinatorial methods.

Abstract

In 1966, Hedetniemi conjectured that for any positive integer $n$ and graphs $G$ and $H$, if neither $G$ nor $H$ is $n$-colourable, then $G \times H$ is not $n$-colourable. This conjecture has received significant attention over the past half century, and was disproved by Shitov in 2019. Shitov's proof shows that Hedetniemi's conjecture fails for sufficiently large $n$. Shortly after Shitov's result, smaller counterexamples were found in a series of papers, and it is now known that Hedetniemi's conjecture fails for all $n \ge 4$, and holds for $n \le 3$. Hedetniemi's conjecture has inspired extensive research, and many related problems remain open. This paper surveys the results and problems associated with the conjecture, and explains the ideas used in finding counterexamples.