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 < Δ ( 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 2 x -regular graph G on n vertices which: 1. Hamilton decomposition, and complement in K n that is primitive. 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.