Triangular embeddings of complete graphs from graceful labellings of paths

Goddyn, Luis; Richter, R. Bruce and Širáň, Jozef (2007). Triangular embeddings of complete graphs from graceful labellings of paths. Journal of Combinatorial Theory, Series B, 97(6) pp. 964–970.

DOI: https://doi.org/10.1016/j.jctb.2007.02.009

Abstract

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 2^2s 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 11^s distinct cyclic oriented triangular embedding of a complete graph of order 12s+7 for all sufficiently large s.

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