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

摘要

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 Delta(G). In this paper, it is shown that chi' (G) = Delta(G) if (1) Delta(G) >= 9 and G contains no adjacent chordal 5-cycles, or (2) Delta(G) >= 8 and G contains no adjacent 4-cycles or no 5-cycles. (C) 2014 Elsevier B.V. All rights reserved.