  We develop the theory of quantum (a.k.a. noncommutative) relations and
quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant
setting, where all systems (finite-dimensional $C^*$-algebras) carry an action
of a compact quantum group $G$, and all channels (completely positive maps
preserving the canonical $G$-invariant state) are covariant with respect to the
$G$-actions. We motivate our definitions by applications to zero-error quantum
communication theory with a symmetry constraint. Some key results are the
following: 1) We give a necessary and sufficient condition for a covariant
quantum relation to be the underlying relation of a covariant channel. 2) We
show that every quantum confusability graph with a $G$-action (which we call a
quantum $G$-graph) arises as the confusability graph of a covariant channel. 3)
We show that a covariant channel is reversible precisely when its confusability
$G$-graph is discrete. 4) When $G$ is quasitriangular (this includes all
compact groups), we show that covariant zero-error source-channel coding
schemes are classified by covariant homomorphisms between confusability
$G$-graphs.
