Hedetniemi's Conjecture and the Retracts of a Product of Graphs

We show that every core graph with a primitive automorphism group has the property that whenever it is a retract of a product of connected graphs, it is a retract of a factor. The example of Kneser graphs shows that the hypothesis that the factors are connected is essential. In the case of complete graphs, our result has already been shown in [4, 17], and it is an instance where Hedetniemi’s conjecture is known to hold. In fact, our work is motivated by a reinterpretation of Hedetniemi’s conjecture in terms of products and retracts.