Symmetric Hamilton cycle decompositions of complete multigraphs

Let n ≥ 3 and ⋋ ≥ 1 be integers. Let ⋋Kn denote the complete multigraph with edge-multiplicity ⋋. In this paper, we show that there exists a symmetric Hamilton cycle decomposition of ⋋K2m for all even ⋋ ≥ 2 and m ≥ 2. Also we show that there exists a symmetric Hamilton cycle decomposition of ⋋K2m − F for all odd ⋋ ≥ 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 ⋋Kn (respectively, ⋋Kn − F, where F is a 1-factor of ⋋Kn) which exist if and only if ⋋(n − 1) is even (respectively, ⋋(n − 1) is odd), except the non-existence cases n ≡ 0 or 6 (mod 8) when ⋋ = 1