Multigraph Overfull Conjecture

Let $G$ be a multigraph with no loops, and let $\mu(G)$ be its maximum edge multiplicity. Let $\Delta(G)$ be its maximum degree, let $\chi'(G)$ be its chromatic index, and call a subgraph $H$ $\Delta(G)$-overfull when $\Delta(H)=\Delta(G)$ and $|E(H)|>\Delta(H)\lfloor |V(H)|/2\rfloor$.

Multigraph Overfull Conjecture. If

$$\Delta(G)>\frac{1}{3}\mu(G)|V(G)|,$$

then $\chi'(G)=\Delta(G)$ if and only if $G$ contains no $\Delta(G)$-overfull subgraph.

This is the multigraph analogue of the Overfull Conjecture. The degree condition is shown in the source to be best possible using suitable multigraphs derived from the Petersen graph, but the conjecture itself is not resolved there.