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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics