Abstract
A vector t-coloring of a graph is an assignment of real vectors p1, ... , pn to its vertices such that piTpi=t-1, for all i= 1 , ... , n and piTpj= 1 for which a vector t-coloring of G exists. For a graph H and a vector t-coloring p1, ... , pn of G, the map taking (i, ℓ) ELEMENT OF V(G) x V(H) to pi is a vector t-coloring of the categorical product Gx H.
It follows that the vector chromatic number of Gx H is at most the minimum of the vector chromatic numbers of the factors. We prove that equality always holds, constituting a vector coloring analog of the famous Hedetniemi Conjecture from graph coloring.
Furthermore, we prove necessary and sufficient conditions under which all optimal vector colorings of Gx H are induced by optimal vector colorings of the factors. Our proofs rely on various semidefinite programming formulations of the vector chromatic number and a theory of optimal vector colorings we develop along the way, which is of independent interest.