... 1-factors of G. This gives a decomposition of the edges of G into 2r 1-factors of G,a(k + 1) -regular simple graph, and (r − 1) k -regular simple graphs.3 1-factorization of regular multigraphs of even ... ,r2 are Hamilton cycles of G, F1,F2, ,F(r−1) are 1-factors of G, F is a simple (k +1)-factor of G,andS1,S2, ,S(r−1) are k -regular simplesubgraphs of G.Sincen is even, each of the ... subgraphF of G such that degF(v)=f (v) for each v ∈ V (G). For X, Y ⊆ V (G)wedenotethe electronic journal of combinatorics 8 (2001), #R41 3 All regular multigraphs of even order and high degree are...