  We investigate vector chromatic number, Lovasz theta of the complement, and
quantum chromatic number from the perspective of graph homomorphisms. We prove
an analog of Sabidussi's theorem for each of these parameters, i.e. that for
each of the parameters, the value on the Cartesian product of graphs is equal
to the maximum of the values on the factors. We also prove an analog of
Hedetniemi's conjecture for Lovasz theta of the complement, i.e. that its value
on the categorical product of graphs is equal to the minimum of its values on
the factors. We conjecture that the analogous results hold for vector and
quantum chromatic number, and we prove that this is the case for some special
classes of graphs.
