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) = maxx 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 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