## Abstract

The two diagrams in Figure 8.1 represent the same graph, a

*K*_{4}with vertices*a, b, c*and*d*. As diagrams they are quite different: in the version*K*_{4}(1), the edges*ac*and*bd*cross; in*K*_{4}(2) there are no crossings. We shall refer to the two diagrams as different*representations*of the graph in the plane. The*crossing number*of a representation is the number of different pairs of edges that cross; the crossing number ν(*G*) of a graph*G*is the minimum number of crossings in any representation of*G*. A representation is called*planar*if it contains no crossings, and a*planar graph*is a graph that has a planar representation. In other words, a*planar graph G*is one for which ν(*G*) = 0. Figure 8.1 shows that ν(*K*_{4}) = 0.## Keywords

Planar Graph Complete Graph Exchange Color Complete Bipartite Graph Plane Representation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Second Edition, Birkhäuser Boston 2007