Some Cyclic Solutions to the Three Table Oberwolfach Problem

We use graceful labellings of paths to give a new way of constructing terraces for cyclic groups. These terraces are then used to find cyclic solutions to the three table Oberwolfach problem, OP(r, r, s), where two of the tables have equal size. In particular we show that, for every odd r ≥ 3 and even r with 4 ≤ r ≤ 16, there is a number Nr such that there is a cyclic solution to OP(r, r, s) whenever s ≥ Nr. The terraces we are able to construct also prove a conjecture of Anderson: For all m ≥ 3, there is a terrace of Z2m which begins 0, 2k, k, . . . for some k.