The problem
For every multigraph \(G\), the edge-chromatic number satisfies \(\chi'(G) \le \max(\Delta(G)+1, \Gamma(G))\), where \(\Gamma\) is the fractional lower bound. (Resolved 2019.)
The Goldberg–Seymour conjecture
historicHistory & significance
Goldberg (1973) and Seymour (1979) independently conjectured the fractional edge-chromatic number always rounds up by less than one — the exact form of the gap between fractional and integral edge colouring. Partial results accumulated for decades (Tashkinov trees, Kahn's asymptotics), but the full statement resisted until Chen–Jing–Zang announced a proof in 2019, built on an extended Tashkinov-tree and discharging machinery of formidable size.
Connected problems
- related The total colouring conjecture
More in Graph theory
- same area The strong perfect graph theorem
- same area The Petersen colouring conjecture
- same area The total colouring conjecture
References
- primaryRead the primary source →
- sourceGoldberg–Seymour conjecture
The resolution (human proof)
Chen–Jing–Zang (announced 2019) proved χ′(G) ≤ max(Δ+1, Γ(G)) for every multigraph — the fractional chromatic index always determines the integral one up to the unavoidable +1 — via extended Tashkinov trees and large-scale discharging.
Verified by curator — see the API for full claim provenance.