  Let $(\Gamma,\gamma)$ be a crystallization of connected compact 3-manifold
$M$ with $h$ boundary components. Let $\mathcal{G}(M)$ and $\mathit k (M)$ be
the regular genus and gem-complexity of $M$ respectively, and let
$\mathcal{G}(\partial M)$ be the regular genus of $\partial M$. We prove that
  $$\mathit k (M)\geq 3 (\mathcal{G}(M)+h-1) \geq 3 (\mathcal{G} (\partial
M)+h-1).$$ These bounds for gem-complexity of $M$ are sharp for several
3-manifolds with boundary. Further, we show that if $\partial M$ is connected
and $\mathit k (M)< 3 (\mathcal{G} (\partial M)+1)$ then $M$ is a handlebody.
In particular, we prove that $\mathit k (M) =3 \mathcal{G} (\partial M)$ if $M$
is a handlebody and $\mathit k (M) \geq 3 (\mathcal{G} (\partial M)+1)$ if $M$
is not a handlebody. Further, we obtain several combinatorial properties for a
crystallization of 3-manifolds with boundary.
