Every 4-colorable graph with maximum degree 4 has an equitable 4-coloring

Chen et al., conjectured that for r≥3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are K_{r, r} (for odd r) and K_{r + 1}. If true, this would be a joint strengthening of the Hajnal-Szemerédi theorem and Brooks' theorem. Chen et al., proved that their conjecture holds for r = 3. In this article we study properties of the hypothetical minimum counter-examples to this conjecture and the structure of "optimal" colorings of such graphs. Using these properties and structure, we show that the Chen-Lih-Wu Conjecture holds for r≤4.