Vector coloring the categorical product of graphs

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, 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 p1, … , pn of G, the map taking (i, ℓ) ∈ V(G) × V(H) to pi is a vector t-coloring of the categorical product G× H. It follows that the vector chromatic number of G× 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× 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.