Graphs Satisfying a Richness Condition and One Concerning Forbidden Subgraphs

We characterize several classes of graphs which satify a richness condition (R) and a condition (F) which concerns forbidden subgraphs. In particular we treat the case where (R) is: Each two vertices have a common neighbor. And this in combination with (F): G has no 5-circle C ₅ resp. no 6-circle C ₆ as subgraph. Another result deals with the case where (R) and (F) concern existence or non-existence of circles C _n .