Indecomposable 1‐factorizations of the complete multigraph for every

A 1‐factorization of the complete multigraph is said to be indecomposable if it cannot be represented as the union of 1‐factorizations of and , where . It is said to be simple if no 1‐factor is repeated. For every and for every , we construct an indecomposable 1‐factorization of , which is not simple. These 1‐factorizations provide simple and indecomposable 1‐factorizations of for every and . We also give a generalization of a result by Colbourn et al., which provides a simple and indecomposable 1‐factorization of , where , , p prime.