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Keywords:

  • complete graph;
  • complete bipartite graph;
  • 1-factor;
  • Hamilton cycle;
  • Hamilton cycle decomposition;
  • symmetric

Abstract

Let n≥2 be an integer. The complete graph Kn with a 1-factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that KnF has a decomposition into Hamilton cycles which are symmetric with respect to the 1-factor F if and only if n≡2, 4 mod 8. We also show that the complete bipartite graph Kn, n has a symmetric Hamilton cycle decomposition if and only if n is even, and that if F is a 1-factor of Kn, n, then Kn, nF has a symmetric Hamilton cycle decomposition if and only if n is odd. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:1-15, 2010