Relatively small counterexamples to Hedetniemi's conjecture

Abstract

Hedetniemi conjectured in 1966 that χ(G×H)=min⁡{χ(G),χ(H)} for all graphs G and H. Here G×H is the graph with vertex set V(G)×V(H) defined by putting (x,y) and (x′,y′) adjacent if and only if xx′∈E(G) and yy′∈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 p33p. In this paper, we show that the conjecture fails already for some graphs G and H with chromatic number 3⌈p+12⌉ and with p⌈(p−1)/2⌉ and 3⌈p+12⌉(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.