We propose a connection-based characterization of the multiplicative fragment of non-commutative logic (MNL), that is a conservative extension of both commutative (MLL) and non-commutative or cyclic (MCyLL) linear logic. It is based on a characterization for MLL together with the introduction of labels and constraints from a formula polarization process. We also study a similar characterization for the intuitionistic fragment of MNL. Finally, we consider the relationships between these results and proof nets construction in MNL based on labels and constraints.