A homomorphism from a graph X to a graph Y is an adjacency preserving

mapping f:V(X) -> V(Y). We consider a nonlocal game in which Alice and

Bob are trying to convince a verifier with certainty that a graph X

admits a homomorphism to Y. This is a generalization of the

well-studied graph coloring game. Via systematic study of quantum

homomorphisms we prove new results for graph coloring. Most

importantly, we show that the Lovász theta number of the complement lower bounds the quantum chromatic number, which itself is not known to be computable. We also show that other quantum graph parameters, such as quantum independence number, can differ from their classical counterparts. Finally, we show that quantum homomorphisms closely relate to zero-error channel capacity. In particular, we use quantum

homomorphisms to construct graphs for which entanglement-assistance

increases their one-shot zero-error capacity.