  We present the Darboux transformations for a novel class of two-dimensional
discrete integrable systems named as $\mathbb{Z}_\mathcal{N}$ graded discrete
integrable systems, which were firstly proposed by Fordy and Xenitidis within
the framework of $\mathbb{Z}_\mathcal{N}$ graded discrete Lax pairs very
recently. In this paper, the $\mathbb{Z}_\mathcal{N}$ graded discrete equations
in coprime case and their corresponding Lax pairs are derived from the discrete
Gel'fand-Dikii hierarchy by applying a transformation of the independent
variables. The construction of the Darboux tranformations is realised by
considering the associated linear problems in the bilinear formalism for the
$\mathbb{Z}_\mathcal{N}$ graded lattice equations. We show that all these
$\mathbb{Z}_\mathcal{N}$ graded equations share a unified solution structure in
our scheme.
