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

  • Authors:

  • Henry A. Kierstead;Alexandr V. Kostochka

  • Affiliations:

  • Arizona State University, School of Mathematical and Statistical Sciences, P.O. Box 871804, 85287, Tempe, AZ, USA;University of Illinois, Department of Mathematics, 1409 W. Green St., 61801, Urbana, IL, USA and Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

  • Venue:
  • Combinatorica
  • Year:
  • 2010

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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 ≤ i ≤ t, 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.