Symmetric Hamilton cycle decompositions of complete graphs minus a 1-factor

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 Kn−F 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, n−F has a symmetric Hamilton cycle decomposition if and only if n is odd. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:1-15, 2010