Sabidussi versus Hedetniemi for three variations of the chromatic number

Abstract

We investigate vector chromatic number (źvec), Lovasz V-function of the complement $$(\bar \vartheta )$$, and quantum chromatic number (źq) 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. Interestingly, as a consequence of this result for $$\bar \vartheta$$, we obtain analog of Hedetniemi's conjecture, i.e., that the value of $$\bar \vartheta$$ 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.