  Handsome proof nets were introduced by Retor\'e as a syntax for
multiplicative linear logic. These proof nets are defined by means of cographs
(graphs representing formulas) equipped with a vertices partition satisfying
simple topological conditions. In this paper we extend this syntax to
multiplicative linear logic with units and exponentials. For this purpose we
develop a new sound and complete sequent system for the logic, enforcing a
stronger notion of proof equivalence with respect to the one usually considered
in the literature. We then define combinatorial proofs, a graphical proof
system able to capture syntactically the proof equivalence, for the cut-free
fragment of the calculus. We conclude the paper by defining the exponentially
handsome proof nets as combinatorial proofs with cuts and defining an internal
normalization procedure for this syntax.
