### Abstract

Corrádi and Hajnal proved that for every k ≥ 1 and n ≥ 3k, every n-vertex graph with minimum degree at least 2k contains k vertex-disjoint cycles. This implies that every 3k-vertex graph with maximum degree at most k − 1 has an equitable k-coloring. We prove that for s∈{3,4} if an sk-vertex graph G with maximum degree at most k has no equitable k-coloring, then G either contains K<inf>k+1</inf> or k is odd and G contains K<inf>k,k</inf>. This refines the above corollary of the Corrádi-Hajnal Theorem and also is a step toward the conjecture by Chen, Lih, and Wu that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are K<inf>r,r</inf> (for odd r) and K<inf>r+1</inf>.

Original language | English (US) |
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Pages (from-to) | 497-512 |

Number of pages | 16 |

Journal | Combinatorica |

Volume | 35 |

Issue number | 4 |

DOIs | |

State | Published - Aug 5 2014 |

### Keywords

- 05C15
- 05C35

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*35*(4), 497-512. https://doi.org/10.1007/s00493-014-3059-6