### Abstract

The Ore-degree of an edge xy in a graph G is the sum θ (x y) = d (x) + d (y) of the degrees of its ends. In this paper we discuss colorings and equitable colorings of graphs with bounded maximum Ore-degree, θ (G) = max_{x y ∈ E (G)} θ (x y). We prove a Brooks-type bound on chromatic number of graphs G with θ (G) ≥ 12. We also discuss equitable and nearly equitable colorings of graphs with bounded maximum Ore-degree: we characterize r-colorable graphs with maximum Ore-degree at most 2r whose every r-coloring is equitable. Based on this characterization, we pose a conjecture on equitable r-colorings of graphs with maximum Ore-degree at most 2r, which extends the Chen-Lih-Wu Conjecture and one of our earlier conjectures. We prove that our conjecture is true for r = 3.

Original language | English (US) |
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Pages (from-to) | 298-305 |

Number of pages | 8 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 99 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2009 |

### Keywords

- Brooks' theorem
- Edge degree
- Equitable coloring
- Graph coloring

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*99*(2), 298-305. https://doi.org/10.1016/j.jctb.2008.06.007