Vector coloring the categorical product of graphs

A vector t -coloring of a graph is an assignment of real vectors $$p_1, \ldots , p_n$$ p 1 , … , p n to its vertices such that $$p_i^Tp_i = t-1,$$ p i T p i = t - 1 , for all $$i=1, \ldots , n$$ i = 1 , … , n and $$p_i^Tp_j \le -1$$ p i T p j ≤ - 1 , whenever i and j are adjacent. The vector chromatic number of G is the smallest number $$t \ge 1$$ t ≥ 1 for which a vector t -coloring of G exists. For a graph H and a vector t -coloring $$p_1,\ldots ,p_n$$ p 1 , … , p n of G , the map taking $$(i,\ell )\in V(G)\times V(H)$$ ( i , ℓ ) ∈ V ( G ) × V ( H ) to $$p_i$$ p i is a vector t -coloring of the categorical product $$G \times H$$ G × H . It follows that the vector chromatic number of $$G \times H$$ 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 $$G\times H$$ 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.