Indecomposable 1‐factorizations of the complete multigraph for every

Abstract

A 1-factorization of the complete multigraph 2 is said to be indecomposable if it cannot be represented as the union of 1-factorizations of 0 2 and ( − 0 ) 2 , where 0 < . It is said to be simple if no 1-factor is repeated. For every ≥ 9 and for every ( − 2)∕3 ≤ ≤ 2 , we construct an indecomposable 1-factorization of 2 , which is not simple. These 1-factorizations provide simple and indecomposable 1-factorizations of 2 for every ≥ 18 and 2 ≤ ≤ 2⌊ ∕2⌋ − 1. We also give a generalization of a result by Colbourn et al., which provides a simple and indecomposable 1-factorization of 2 , where 2 = + 1, = ( − 1)∕2, prime.