Hedetniemi’s Conjecture and Dense Boolean Lattices

Abstract

The category \({\mathcal{D}}\) of finite directed graphs is Cartesian closed, hence it has a product and exponential objects. For a fixed K, let \(K^{\mathcal{D}}\) be the class of all directed graphs of the form K G, preordered by the existence of homomorphisms, and factored by homomorphic equivalence. It has long been known that \(K^{\mathcal{D}}\) is always a Boolean lattice. In this paper we prove that for any complete graph K n with n ≥ 3, \(K_n^{\mathcal{D}}\) is dense, hence up to isomorphism it is the unique countable dense Boolean lattice. In graph theory, the structure of \(K_n^{\mathcal{D}}\) is connected to the conjecture of Hedetniemi on the chromatic number of a categorical product of graphs.