Relatively small counterexamples to Hedetniemi's conjecture

TL;DR

This paper demonstrates that Hedetniemi's conjecture, which relates the chromatic number of graph products to the minimum of individual chromatic numbers, fails for smaller graphs than previously known, with explicit bounds on their size and chromatic number.

Contribution

The paper provides new, smaller counterexamples to Hedetniemi's conjecture, improving upon the size and chromatic number bounds of previously known counterexamples.

Findings

Hedetniemi's conjecture fails for graphs with chromatic number around 225.

Counterexamples exist with significantly fewer vertices than earlier known examples.

Explicit bounds on the size and chromatic number of counterexamples are established.

Abstract

Hedetniemi conjectured in 1966 that $\chi(G \times H) = \min\{\chi(G), \chi(H)\}$ for all graphs $G$ and $H$. Here $G\times H$ is the graph with vertex set $ V(G)\times V(H)$ defined by putting $(x,y)$ and $(x',y')$ adjacent if and only if $xx'\in E(G)$ and $yy'\in E(H)$. This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let $p$ be the minimum number of vertices in a graph of odd girth $7$ and fractional chromatic number greater than $3+4/(p-1)$. Shitov's proof shows that Hedetniemi's conjecture fails for some graphs with chromatic number about $p^22^{p+1} $ and with about $(p^22^{p+1})^{p^32^{p-1}} $ vertices. In this paper, we show that the conjecture fails already for some graphs $G$ and $H$ with chromatic number $3\lceil \frac {p+1}2 \rceil $ and with $p \lceil (p-1)/2 \rceil$ and $3 \lceil \frac {p+1}2 \rceil (p+1)-p$ vertices, respectively. The currently known upper bound for $p$ is $148$. Thus Hedetniemi's conjecture fails for some graphs $G$ and $H$ with chromatic number $225$, and with $10,952$ and $33,377$ vertices, respectively.