  We present two lists of multi-component systems of integrable difference
equations defined on the edges of a $\mathbb{Z}^2$ graph. The integrability of
these systems is manifested by their Lax formulation which is a consequence of
the multi-dimensional compatibility of these systems. Imposing constraints
consistent with the systems of difference equations, we recover known
integrable quad-equations including the discrete version of the
Krichever-Novikov equation. The systems of difference equations allow us for a
straightforward reformulation as Yang-Baxter maps. Certain two-component
systems of equation defined on the vertices of a $\mathbb{Z}^2$ lattice, their
non-potential form and integrable equations defined on 5-point stencils, are
also obtained.
