Ore-type versions of Brooks' theorem

The Ore-degree of an edge xy in a graph G is the sum θ(xy)=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)=maxxy∈E(G)θ(xy). 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.