Minimal crystallizations of 3-manifolds with boundary

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