Symmetric Hamilton cycle decompositions of complete multigraphs

Let n ≥ 3 and λ ≥ 1 be integers. Let λKn denote the complete multigraph with edge-multiplicity λ. In this paper, we show that there exists a symmetric Hamilton cycle decomposition of λK2m for all even λ ≥ 2 and m ≥ 2. Also we show that there exists a symmetric Hamilton cycle decomposition of λK2m − F for all odd λ ≥ 3 and m ≥ 2. In fact, our results together with the earlier results (by Walecki and Brualdi and Schroeder) completely settle the existence of symmetric Hamilton cycle decomposition of λKn (respectively, λKn − F , where F is a 1-factor of λKn) which exist if and only if λ(n − 1) is even (respectively, λ(n − 1) is odd), except the non-existence cases n ≡ 0 or 6 (mod 8) when λ = 1.