An Asymptotic Version of the Multigraph 1-Factorization Conjecture

We give a self-contained proof that for all positive integers $r$ and all $ϵ>0$, there is an integer $N=N(r,ϵ)$ such that for all $n \ge N$anyregularmultigraphoforder$2n$withmultiplicityatmost$r$ and degree at least $(1+ϵ)rn$ is 1-factorizable. This generalizes results of Perkovi{'c} and Reed, and Plantholt and Tipnis.