Relatively small counterexamples to Hedetniemi's conjecture

Hedetniemi conjectured in 1966 that chi(G x H) = min{chi(G), chi(H)} for all graphs G and H. Here G x H is the graph with vertex set V(G) x V(H) defined by putting (x, y) and (x', y') adjacent if and only if xx' is an element of E(G) and yy' is an element of 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[ p+1 2 1 and with p[(p - 1)/21 and 3[p+1 2 1(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. (c) 2020 Elsevier Inc. All rights reserved.