Triangular embeddings of complete graphs from graceful labellings of paths

Authors:
Luis Goddyn;R. Bruce Richter;Jozef Širáň
Affiliations:
Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada;Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada;Department of Mathematics (SvF), Slovak University of Technology, Bratislava, Slovakia
Venue:
Journal of Combinatorial Theory Series B
Year:
2007

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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 2^2^s 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)^2^s, 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.