Triangular embeddings of complete graphs from graceful labellings of paths

We show that to each graceful labelling of a path on 2 s + 1 vertices, s ⩾ 2 , there corresponds a current assignment on a 3-valent graph which generates at least 2 2 s cyclic oriented triangular embeddings of a complete graph on 12 s + 7 vertices. We also show that in this correspondence, two distinct graceful labellings never give isomorphic oriented embeddings. Since the number of graceful labellings of paths on 2 s + 1 vertices grows asymptotically at least as fast as ( 5 / 3 ) 2 s , this method gives at least 11 s distinct cyclic oriented triangular embedding of a complete graph of order 12 s + 7 for all sufficiently large s.