Computational complexity of some restricted instances of 3-SAT

Tovey [A simplified satisfiability problem, Discrete Appl. Math. 8 (1984) 85–89] showed that it is NP-hard to decide the satisfiability of 3-SAT instances in which every variable occurs four times, while every instance of 3-SAT in which each variable occurs three times is satisfiable. We explore the border between these two problems. Answering a question of