We show that for any n ≥ 13, there exist graphs with chromatic number larger than n whose product has chromatic number n. Our construction is an adaptation of the construction of counterexamples to Hedetniemi’s conjecture devised by Shitov, and adapted by Zhu to graphs with relatively small chromatic numbers. The new tools we introduce are graphs with minimal colourings that are “wide” in the sense of Simonyi and Tardos, and generalised Mycielskians to settle the case n = 13.

中文翻译：

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14色图乘积的色数可以是13

我们证明，对于任何n ≥ 13，存在色数大于n的图，其乘积的色数为n。我们的构造是对 Shitov 设计的 Hedetniemi 猜想的反例构造的改编，并由 Zhu 改编用于具有相对较小色数的图。我们介绍的新工具是具有最小着色的图，它们在 Simonyi 和 Tardos 的意义上是“宽的”，并且广义的 Mycielskiians 解决了n = 13 的情况。