Vector coloring the categorical product of graphs

Abstract

A vector t-coloring of a graph is an assignment of real vectors p(1), ... , p(n) to its vertices such that p(i)(T) p(i) = t - 1, for all i = 1, ... , n and p(i)(T) p(j) <= -1, whenever i and j are adjacent. The vector chromatic number of G is the smallest number t >= 1 for which a vector t-coloring of G exists. For a graph H and a vector t-coloring p(1), ... , p(n) of G, the map taking (i, l) is an element of V(G) x V(H) to p(i) is a vector t-coloring of the categorical product G x H. It follows that the vector chromatic number of G x 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 G x 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.