Indecomposable 1-factorizations of the complete multigraph λK2n for every λ≤2n

A 1-factorization of the complete multigraph λK2n is said to be indecomposable if it cannot be represented as the union of 1-factorizations of λ0K2n and (λ - λ0)K2n, where λ0 < λ. It is said to be simple if no 1-factor is repeated. For every n ≥ 9 and for every (n - 2)/3 ≤ λ ≤ 2n, we construct an indecomposable 1-factorization of λK2n, which is not simple. These 1-factorizations provide simple and indecomposable 1-factorizations of λK2s for every s ≥ 18 and 2 ≤ λ ≤ 2└s/2┘ - 1. We also give a generalization of a result by Colbourn et al., which provides a simple and indecomposable 1-factorization of λK2n, where 2n = pm + 1, λ = (pm - 1)/2, p prime.