  We define vertex-colourings for edge-partitioned digraphs, which unify the
theory of P-partitions and proper vertex-colourings of graphs. We use our
vertex-colourings to define generalized chromatic functions, which merge the
chromatic symmetric and quasisymmetric functions of graphs and generating
functions of P-partitions. Moreover, numerous classical bases of symmetric and
quasisymmetric functions, both in commuting and noncommuting variables, can be
realized as special cases of our generalized chromatic functions. We also
establish product and coproduct formulas for our functions. Additionally, we
construct the new Hopf algebra of r-quasisymmetric functions in noncommuting
variables, and apply our functions to confirm its Hopf structure, and establish
natural bases for it.
