Every graph with n vertices and maximum vertex degree Δ(G)<=k is (k + 1)-colorable with all color classes of size ⌊n/(k + 1)⌋ or ⌈n/(k + 1)⌉, where ⌊x⌋ is the floor function and ⌈x⌉ is the ceiling function.

Seymour conjecture