  In this paper, we develop the perfect basis theory for quantum
Borcherds-Bozec algebras $U_{q}(\mathfrak g)$ and their irreducible highest
weight modules $V(\lambda)$. We show that the lower perfect graph (resp. upper
perfect graph) of every lower perfect basis (resp. upper perfect basis) of
$U_{q}^{-}(\mathfrak g)$ (resp. $V(\lambda)$) is isomorphic to the crystal
$B(\infty)$ (resp. $B(\lambda)$).
