Some sufficient conditions for 1-planar graphs to be Class 1

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)Δ(G). In this paper, it is shown that χ′(G)=Δ(G)χ′(G)=Δ(G) if (1) Δ(G)≥9Δ(G)≥9 and G   contains no adjacent chordal 5-cycles, or (2) Δ(G)≥8Δ(G)≥8 and G contains no adjacent 4-cycles or no 5-cycles.