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

Abstract

AbstractWe show that every core graph with a primitive automorphismgroup has the property that whenever it is a retract of a product ofconnected graphs, it is a retract of a factor. The example of Knesergraphs shows that the hypothesis that the factors are connected isessential. In the case of complete graphs, our result has already beenshown in [4, 17], and it is an instance where Hedetniemi’s conjectureis known to hold. In fact, our work is motivated by a reinterpretationof Hedetniemi’s conjecture in terms of products and retracts. 1 Introduction One of the outstanding problems in graph theory is a formula concerning thechromatic number of a product of graphs:Conjecture 1.1 (Hedetniemi [9]) χ(G×H) = min{χ(G),χ(H)}.Here, G× His the product of Gand H, defined byV(G× H) = V(G)×V(H)E(G× H) = {[(u 1 ,u 2 ),(v 1 ,v 2 )] : [u 1 ,v 1 ] ∈ E(G) and [u 2 ,v 2 ] ∈ E(H)}.A colouring of G× H can be derived from a colouring of any of its fac-tors, hence χ(G× H) ≤ min{χ(G),χ(H)}. The inherent difficulty of Con-jecture 1.1 lies in finding a lower bound for χ(G× H). It is known that1