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

Henry Kierstead, Alexandr V. Kostochka

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

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 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.

Original languageEnglish (US)
Pages (from-to)201-216
Number of pages16
JournalCombinatorica
Volume30
Issue number2
DOIs
StatePublished - Sep 24 2010

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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