Comparing graph databases with traditional,e.g., relational databases, some important database features are often missing there. Particularly, a graph database schema including integrity constraints is mostly not explicitly defined, also a conceptual modelling is not used. It is hard to check a consistency of the graph database, because almost no integrity constraints are defined or only their very simple representatives can be specified. In the paper, we discuss these issues and present current possibilities and challenges in graph database modelling. We focus also on integrity constraints modelling and propose functional dependencies between entity types, which reminds modelling functional dependencies known from relational databases. We show a number of examples of often cited GDBMSs and their approach to database schemas and ICs specification. Also a conceptual level of a graph database design is considered. We propose a sufficient conceptual model based on a binary variant of the ER model and show its relationship to a graph database model, i.e. a mapping conceptual schemas to database schemas. An alternative based on the conceptual functions called attributes is presented. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.