Indecomposable $1$-factorizations of the complete multigraph $\lambda K_{2n}$ for every $\lambda \le 2n$

Abstract

A $1$-factorization of the complete multigraph $\lambda K_{2n}$ is said to be indecomposable if it cannot be represented as the union of $1$-factorizations of ${\lambda }_0{K}_{2n}$ and $(\lambda -{\lambda }_0)K_{2n}$, where ${\lambda }_0<\lambda$. It is said to be simple if no $1$-factor is repeated. For every $n\ge 9$ and for every $(n-2)/3\le \lambda \le 2n$, we construct an indecomposable $1$-factorization of $\lambda K_{2n}$ which is not simple. These $1$-factorizations provide simple and indecomposable $1$-factorizations of $\lambda K_{2s}$ for every $s\ge 18$ and $2\le \lambda \le 2\lfloor s/2\rfloor -1$. We also give a generalization of a result by Colbourn et al. which provides a simple and indecomposable $1$-factorization of $\lambda K_{2n}$, where $2n=p^m+1$, $\lambda =(p^m-1)/2$, $p$ prime.