Goldberg–Seymour conjecture
In graph theory the Goldberg–Seymour conjecture states that
where 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.
Above G can be a multigraph.Background
It is already known that for loopless G :
When does equality not hold? It does not hold for the Petersen graph. It is hard to find other examples. It is currently unknown whether there are any planar graphs for which equality does not hold.
This conjecture is named after Paul Seymour of Princeton University, who arrived to it independently of Goldberg.In 2019, an alleged proof was announced by Chen, Jing, and Zang in the paper. Part of their proof was to find a suitable generalization of Vizing's theorem to multigraphs.
