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 $(x^{\prime },y^{\prime })$ adjacent if and only if $x{x}^{\prime }\in E(G)$ and $y{y}^{\prime }\in E(H)$. 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 $p^3{3}^p$. 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 83. Thus Hedetniemi's conjecture fails for some graphs G and H with chromatic number 126, and with $3,403$ and $10,501$ vertices, respectively.

赫迪涅米（Hedetniemi）在1966年推测 $\chi （G\times H）=\mathrm{分}\{\chi （G），\chi （H）\}$对于所有的图ģ和ħ。这里$G\times H$ 是顶点集的图 $V（G）\times V（H）$ 通过放置定义 $（X，ÿ）$ 和 $（X^{\prime }，{ÿ}^{\prime }）$ 当且仅当 $X{X}^{\prime }\in Ë（G）$ 和 $ÿ{ÿ}^{\prime }\in Ë（H）$。在过去的半个世纪中，这一猜想受到了很多关注。最近，Shitov驳斥了这一猜想。设p为奇数周长7和分数色数大于的图中最小的顶点数$3+4/（p-\mathrm{1个}）$。Shitov的证明表明，Hedetniemi的猜想对于某些色度数约为$p^3{3}^p$。在本文中，我们表明，对于某些色数为G和H的图形，猜想已经失效$3\lceil \frac{p+\mathrm{1个}}{2}\rceil$ 与 $p\lceil （p-\mathrm{1个}）/2\rceil$ 和 $3\lceil \frac{p+\mathrm{1个}}{2}\rceil （p+\mathrm{1个}）-p$顶点。当前已知的p的上限是83。因此，对于某些色度数为126的图形G和H，Hedetniemi的猜想失败了。$3，403$ 和 $10，501$ 顶点。