Triangular embeddings of complete graphs from graceful labellings of paths

We show that to each graceful labelling of a path on 2s+1 vertices, s⩾2, there corresponds a current assignment on a 3-valent graph which generates at least 22s cyclic oriented triangular embeddings of a complete graph on 12s+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 2s+1 vertices grows asymptotically at least as fast as (5/3)2s, this method gives at least s11 distinct cyclic oriented triangular embedding of a complete graph of order 12s+7 for all sufficiently large s.