  In the Constrained-degree percolation model on a graph
$(\mathbb{V},\mathbb{E})$ there are a sequence, $(U_e)_{e\in\mathbb{E}}$, of
i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$.
Each bond $e$ tries to open at time $U_e$, it succeeds if both its end-vertices
would have degrees at most $k-1$. We prove a phase transition theorem for this
model on the square lattice $\mathbb{L}^2$, as well as on the d-ary regular
tree. We also prove that on the square lattice the infinite cluster is unique
in the supercritical phase.
