In Ref. 8, we introduced a simplifying assumption about entity-relationship diagrams (ERDs), called regularity, and showed that regular ERDs have several desirable properties. One such property is that every relation schema in the ERD's canonical relational scheme can be put into Third Normal Form. We left open there the more basic question: under what conditions would the original relation schemas actually be in Boyce-Codd Normal Form (BCNF)? Since the visible semantics of ERDs determine naturally their associated functional dependencies (fd's), it is important to know when an ERD, as designed, already has this strongest normal form given purely in terms of fd's. We show here a sufficient diagrammatic condition (loop-free) under which a regular ERD will have databases enjoying the benefits of BCNF.