Goldberg–Seymour conjecture

In graph theory, the Goldberg–Seymour conjecture states that, for a multigraph $G$ $\operatorname {\chi '} (G)\leq \max(1+\operatorname {\Delta } (G),\,\operatorname {\Gamma } (G))$ where $\operatorname {\chi '} (G)$ is the edge chromatic number of G, $\operatorname {\Delta } (G)$ is its maximum degree, and $\operatorname {\Gamma } G=\max _{H\subset G}{\frac {|E(H)|}{\lfloor {\frac {1}{2}}|V(H)|\rfloor }}.$ This above quantity is twice the arboricity of G. It is sometimes called the density of G. Here, G can be a multigraph and can have loops. For simple graphs, this result follows from Vizing's theorem.