List of Figures vChapter 1. Introduction Chapter 2. Description of 2-crossing-critical graphs with V 10 Chapter 3. Moving into the projective plane Chapter 4. Bridges Chapter 5. Quads have BOD Chapter 6. Green cycles Chapter 7. Exposed spoke with additional attachment not in Q 0 Chapter 8. G embeds with all spokes in M Chapter 9. Parallel edges Chapter 10. Tidiness and global H-bridges Chapter 11. Every rim edge has a colour Chapter 12. Existence of a red edge and its structure Chapter 13. The next red edge and the tile structure Chapter 14. Graphs that are not 3-connected 16.2. The number of bridges is bounded

If G is a graph that minimally does not embed in a nonorientable surface, then each edge of G is in a subdivision of either K3,3 or K5. However, there is an example of a graph that minimally does not embed in the torus and some edge is in no subdivision of either K3.3 or K5. © 1996 John Wiley & Sons, Inc.

Since the crossing number of K 12 is now known to be 150, it is well-known that simple counting arguments and Kleitman's parity theorem for the crossing number of K 2n+1 combine with a specific drawing of K 13 to show that the crossing number of K 13 is one of the numbers in {217, 219, 221, 223, 225}. We show that the crossing number is not 217.Recently, Pan and Richter [4] used a computer to prove that the crossing number of K 11 is indeed 100. A simple counting argument shows that cr(K 12 ) = 150 (as long as the conjecture holds for K 2n−1 , it automatically holds for K 2n ). The next case is K 13 .

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