Hamilton decompositions of graphs with primitive complements

A k-factor of a graph is a k-regular spanning subgraph. A Hamilton cycle is a connected 2-factor. A graph G is said to be primitive if it contains no k-factor with 1 ≤ k < ∆(G). A Hamilton decomposition of a graph G is a partition of the edges of G into sets, each of which induces a Hamilton cycle. In this paper, by using the amalgamation technique, we find necessary and sufficient conditions for the existence of a 2x-regular graph G on n vertices which: 1. has a Hamilton decomposition, and 2. has a complement in Kn that is primitive. This extends the conditions studied by Hoffman, Rodger, and Rosa [D.G. Hoffman, C.A. Rodger, A. Rosa, Maximal sets of 2-factors and Hamiltonian cycles, J. Combin. Theory Ser. B 57 (1) (1993) 69–76] who considered maximal sets of Hamilton cycles and 2-factors. It also sheds light on construction approaches to the Hamilton–Waterloo problem. © 2008 Elsevier B.V. All rights reserved.