Relatively small counterexamples to Hedetniemi's conjecture

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.