Quantum advantage in zero-error function computation with side information

We consider the problem of zero-error function computation with side information. Alice has a source X and Bob has correlated source Y and they can communicate via either classical or a quantum channel. Bob wants to calculate f(X,Y) with zero error. We aim to characterize the minimum amount of information that Alice needs to send to Bob for this to happen with zero-error. In the classical setting, this quantity depends on the asymptotic growth of χ(G^(m)), the chromatic number of an appropriately defined m-instance "confusion graph". In this work we present structural characterizations of G^(m) and demonstrate two function computation scenarios that have the same single-instance confusion graph. However, in one case there a strict advantage in using quantum transmission as against classical transmission, whereas there is no such advantage in the other case.