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

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Abstract

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 K r,r (for odd r) and K r+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 K r,r and V(G) partitions into subsets V 0, …, V t such that G[V 0] = K r,r and for each 1 ≤ it, G[V i ] = K r . 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.

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Correspondence to Henry A. Kierstead.

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Research of this author is supported in part by the NSA grant MDA 904-03-1-0007.

Research of this author is supported in part by the NSF grant DMS-0650784 and by grant 06-01-00694 of the Russian Foundation for Basic Research.

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Kierstead, H.A., Kostochka, A.V. Equitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture. Combinatorica 30, 201–216 (2010). https://doi.org/10.1007/s00493-010-2420-7

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  • DOI: https://doi.org/10.1007/s00493-010-2420-7

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