An Asymptotic Version of the Multigraph 1-Factorization Conjecture

We give a self-contained proof that for all positive integers \(r\) and all \(\epsilon > 0\), there is an integer \(N = N(r, \epsilon)\) such that for all \(n \ge N\) any regular multigraph of order \(2n\) with multiplicity at most \(r\) and degree at least \((1+\epsilon)rn\) is 1-factorizable. This generalizes results of Perkovi{ć} and Reed, and Plantholt and Tipnis.