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 Kr, r (for odd r) and Kr + 1. If true, this would be a joint strengthening of the Hajnal–Szemeredi 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. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:31–48, 2012 © 2012 Wiley Periodicals, Inc.

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