Hedetniemi猜想的相对小的反例
Relatively small counterexamples to Hedetniemi's conjecture
摘要：
Hedetniemi在1966年证明了$χ(G \times H) = \min\{χ(G), χ(H)\}$对所有的图$G$和$H$。这里$G\times H$是顶点集$ V(G)\times V(H)$的图，通过将$(x,y)$和$(x',y')$相邻来定义当且仅当$xx'\in E(G)$和$yy'\in E(H)$。这个猜想在过去的半个世纪里受到了很多关注。最近，希托夫驳斥了这一猜想。设$p$是具有奇数围长$7$且分数色数大于$3+4/(p-1)$的图的最小顶点数。Shitov的证明表明Hedetniemi猜想对某些图的色数约为$p^22^{p+1} $且顶点数约为$(p^22^{p+1})^{p^32^{p-1}} $的图是不成立的.在本文中，我们证明了猜想已经失败的一些图$G$和$H$的色数$3\lceil \frac {p+1}2 \rceil $和$p \lceil (p-1)/2 \rceil$和$3 \lceil \frac {p+1}2 \rceil (p+1)-p$顶点，分别。目前已知的$p$的上限是$148$。因此，Hedetniemi的猜想对某些图$G$和$H$是不成立的，这些图的色数分别是$225$，顶点分别是$10,952$和$33,377$。
英文摘要：
Hedetniemi conjectured in 1966 that $χ(G \times H) = \min\{χ(G), χ(H)\}$ 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.
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