  We recently introduced a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs
and studied the associated discrete integrable systems (lattice equations). In
this paper we introduce the corresponding Yang-Baxter maps. Many well known
examples belong to this scheme for $N=2$, so, for $N\geq 3$, our systems may be
regarded as generalisations of these.
  In particular, for each $N$ we introduce a generalisation of the map
$H_{III}^B$ in the classification of scalar Yang-Baxter maps. For $N=3$ this is
equivalent to the Yang-Baxter map associated with the discrete modified
Boussinesq equation. For $N\geq 5$ (and odd) we introduce a new family of
Yang-Baxter maps, which have no lower dimensional analogue. We also present
multi-component versions of the Yang-Baxter maps $F_{IV}$ and $F_V$ (given in
the ABS classification).
