Equitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture

Chen, Lih, and Wu conjectured that for r≥3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are Kr,r (for odd r) and Kr+1. If true, this would be a strengthening of the Hajnal–Szemerédi Theorem and Brooks’ Theorem. We extend their conjecture to disconnected graphs. For r≥6 the conjecture says the following: If an r-colorable graph G with maximum degree r is not equitably r-colorable then r is odd, G contains Kr,r and V (G) partitions into subsets V0, . . . ,Vt such that G[V0] =Kr,r and for each 1 ≤ i ≤ t, G[Vi] = Kr. We characterize graphs satisfying the conclusion of our conjecture for all r and use the characterization to prove that the two conjectures are equivalent. This new conjecture may help to prove the Chen–Lih–Wu Conjecture by induction.