# Hedetniemi's conjecture is asymptotically false

## TL;DR

This paper demonstrates that Hedetniemi's conjecture, which relates the chromatic number of graph tensor products, is asymptotically false by constructing graphs with a significantly smaller tensor product chromatic number than expected.

## Contribution

It proves the existence of graph pairs with high chromatic numbers whose tensor product has a much lower chromatic number, disproving Hedetniemi's conjecture asymptotically.

## Key findings

- Existence of graphs with chromatic number at least (1+δ)c
- Tensor product chromatic number can be a constant factor smaller
- Disproof of Hedetniemi's conjecture asymptotically

## Abstract

Extending a recent breakthrough of Shitov, we prove that the chromatic number of the tensor product of two graphs can be a constant factor smaller than the minimum chromatic number of the two graphs. More precisely, we prove that there exists an absolute constant $\delta>0$ such that for all $c$ sufficiently large, there exist graphs $G$ and $H$ with chromatic number at least $(1+\delta)c$ for which $\chi(G \times H) \le c$.