is the edge chromatic number of G and Note this above quantity is twice the arboricity of G. It is sometimes called the density of G.[2] Above G can be a multigraph (can have loops).
This conjecture is named after Mark K. Goldberg of Rensselaer Polytechnic Institute[4] and Paul Seymour of Princeton University, who arrived to it independently of Goldberg.
[3] In 2019, an alleged proof was announced by Chen, Jing, and Zang in the paper.
[3] Part of their proof was to find a suitable generalization of Vizing's theorem (which says that for simple graphs
In 2023, Jing[5] announced a new proof with a polynomial-time edge coloring algorithm achieving the conjectured bound.