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 2 r 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 2 r, which extends the Chen–Lih–Wu Conjecture and one of our earlier conjectures. We prove that our conjecture is true for r = 3 .
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