1. Mathematics Combinatorics Graph Theory Topological Graph Theory David Craft

3. A graph is a set of vertices (or points) together with a set of vertex-pairs called edges.

4. A graph is a set of vertices (or points) together with a set of vertex-pairs called edges. A graph is a set of vertices (or points) together with a set of vertex-pairs called edges. Graph Theory is the study of graphs.

5. An imbedding or embedding (or proper drawing) of a graph is one in which edges do not cross.

6. An imbedding or embedding (or proper drawing) of a graph is one in which edges do not cross. NOT an imbedding

7. An imbedding or embedding (or proper drawing) of a graph is one in which edges do not cross. NOT an imbeddingAn imbedding

8. An imbedding or embedding (or proper drawing) of a graph is one in which edges do not cross. NOT an imbeddingAn imbedding Topological Graph Theory is the study of imbeddings of graphs in various surfaces or spaces

9. Orientable surfaces (without boundary): sphere S 0

10. Orientable surfaces (without boundary): sphere S 0 torus S 1 Orientable surfaces (without boundary): sphere S 0 torus S 1

11. Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2 Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2

12. Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2 n-torus S n Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2 n-torus S n

13. Orientable surfaces (without boundary): sphere S 0 torus S 1 2-torus S 2 n-torus S n The surface S n is said to have genus n

14. Some graphs cannot be imbedded in the sphere… ? ? ?

15. ? ? …but all can be imbedded in in a surface of high enough genus.

16. The main problem in topological graph theory: Given a graph G, determine the smallest genus n so that G imbeds in S n.

17. The main problem in topological graph theory: Given a graph G, determine the smallest genus n so that G imbeds in S n. For G =the answer is n = 1.

18. The main problem in topological graph theory: Given a graph G, determine the smallest genus n so that G imbeds in S n. For G =the answer is n = 1. For G = the answer is n = 3.