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

Let n ≥2 be an integer. The complete graph K n with a 1‐factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that K n − 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 K n, n has a symmetric Hamilton cycle decomposition if and only if n is even, and that if F is a 1‐factor of K n, n , then K n, 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