Equitable Coloring and the Maximum Degree

Let Δ(G) denote the maximum degree of a graph G. The equitable Δ-coloring conjecture asserts that a connected graph G is equitably Δ(G)-colorable if it is different from Km , C2m + 1 and K2m + 1,2m + 1 for all m ⩾ 1. This conjecture is established for graphs G satisfying Δ(G) ⩾ |G|/2 or Δ(G) ⩽ 3.