## 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 left-hand version, the edges *ac* and *bd* cross; in the right-hand version 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 *v*(*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 *v*(*G*) = 0. Figure 8.1 shows that *v*(*K* _{4}) = 0.

## Keywords

Planar Graph Exchange Color Complete Bipartite Graph Plane Representation Petersen Graph## Preview

Unable to display preview. Download preview PDF.