Plantholt–Tipnis multigraph 1-factorization conjecture

Let $G$ be a regular multigraph of order $2n$, and let its multiplicity be at most $r$, meaning that no pair of vertices is joined by more than $r$ edges. A 1-factorization of $G$ is a decomposition of its edge set into perfect matchings.

Plantholt–Tipnis multigraph 1-factorization conjecture. If the degree of $G$ is at least $rn$, then $G$ is 1-factorizable.

This is the bounded-multiplicity multigraph generalization of the Chetwynd–Hilton conjecture. The source paper proves an asymptotic version: for every positive integer $r$ and $\frac{}{}\epsilon>0$, the assertion holds for all sufficiently large $n$ when the degree is at least $(1+\epsilon)rn$; the exact threshold $rn$ remains open.