  A homomorphism from a graph $X$ to a graph $Y$ is an adjacency preserving
mapping $f:V(X) \rightarrow V(Y)$. 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 Lov\'{a}sz theta
number of the complement lower bounds the quantum chromatic number, which
itself is not known to be computable. We also show that some of our newly
introduced graph parameters, namely quantum independence and clique numbers,
can differ from their classical counterparts while others, namely quantum odd
girth, cannot. 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.
