Abstract
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. Let G be a 1-planar graph with maximum degree Δ(G). In this paper, it is shown that χ′(G)=Δ(G) if (1) Δ(G)≥9 and G contains no adjacent chordal 5-cycles, or (2) Δ(G)≥8 and G contains no adjacent 4-cycles or no 5-cycles.
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