On the Quantum Chromatic Numbers of Small Graphs

We make two contributions pertaining to the study of the quantum chromatic numbers of small graphs. Firstly, in an elegant paper, Mancinska and Roberson [Baltic Journal on Modern Computing, 4(4), 846-859, 2016] gave an example of a graph G14 on 14 vertices with quantum chromatic number 4 and classical chromatic number 5, and conjectured that this is the smallest graph exhibiting a separation between the two parameters, as measured by the number of vertices. We describe a computer-assisted proof of this conjecture, thereby resolving a longstanding open problem in quantum graph theory. Our second contribution pertains to the study of the rank-r quantum chromatic numbers. While it can now be shown that for every r, chi q and chi(r) q are distinct, few small examples of separations between these parameters are known. We give the smallest known example of such a separation in the form of a graph G21 on 21 vertices with chi q(G21) = chi(2) q(G21) = 4 and xi(G21) = chi(1) q (G21) = chi(G21) = 5. The previous record was held by a graph Gmsg on 57 vertices that was first considered in the aforementioned paper of Mancinska and Roberson and which satisfies chi q(Gmsg) = 3 and chi(1) q (Gmsg) = 4. In addition, G21 provides the first provable separation between the parameters chi(1) q and chi(2) q . We believe that our techniques for constructing G21 and lower bounding its orthogonal rank could be of independent interest.