The Chromatic Number of the Product of 14-Chromatic Graphs Can BE 13

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.