Even embeddings of the complete graphs and the cycle parities

1. Even embeddings of the complete graphs and the cycle parities Kenta Noguchi Keio University Japan Cycles in Graphs

2. Outline • Definitions • The minimum genus even embeddings • Cycle parities • Rotation systems and current graphs • Problems and main theorems Cycles in Graphs

3. Outline • Definitions • The minimum genus even embeddings • Cycle parities • Rotation systems and current graphs • Problems and main theorems Cycles in Graphs

4. Definitions • : the complete graph on vertices • : orientable surface of genus • : nonorientable surface of genus • : the Euler characteristic of • Embedding of on : drawn on without edge crossings • An evenembedding : an embedding which has no odd faces Cycles in Graphs

5. Outline • Definitions • The minimum genus even embeddings • Cycle parities • Rotation systems and current graphs • Problems and main theorems Cycles in Graphs

6. The minimum genus even embeddings of Theorem A (Hartsfield) The complete graph on vertices can be even embedded on closed surface with Euler characteristic which satisfies following inequality. Cycles in Graphs

7. Cycle parities • In even embedded graphs, the parities of the lengths of homotopiccycles are the same. • Then, even embedded graphs can be classified into several types by parities of lengths of their cycles. Cycles in Graphs

8. A list of the cycle parity Cycles in Graphs

9. Main theorem Theorem 1 For any of the types A, B and C, there is a minimum genus even embedding of if except the case and type C. For any of the types D, E and F, there is a minimum genus even embedding of if except the case and type D. 2012/5/30 Cycles in Graphs Cycles in Graphs 9

10. Outline • Definitions • The minimum genus embeddings • Cycle parities • Rotation systems and current graphs • Problems and main theorems Cycles in Graphs

11. Definition of cycle parities • : a closed surface • : the fundamental group on • a cycle parity of : a homomorphism from simple closed curves on to the parities of their length of Cycles in Graphs

12. Equivalence relation • Let be embedding and be embedding ,then homeomorphism s.t. • On each , we want to count the number of equivalence classes of cycle parities. Cycles in Graphs

13. Equivalence classes on • Trivial (bipartite) • Nontrivial (nonbipartite) Cycles in Graphs

14. Equivalence classes on • Trivial • Type A • Type B • Type C Theorem (Nakamoto, Negami, Ota) G : locally bipartitegraph if G is in type A if G is in type B if G is in type C Cycles in Graphs

15. Equivalence classes on • Trivial • Type D • Type E • Type F Theorem (Nakamoto, Negami, Ota) G : locally bipartite graph if G is in type D if G is in type E if G is in type F Cycles in Graphs

16. Main theorem Theorem 1 For any of the types A, B and C, there is a minimum genus even embedding of if except the case and type C. For any of the types D, E and F, there is a minimum genus even embedding of if except the case and type D. 2012/5/30 Cycles in Graphs Cycles in Graphs 16

17. Outline • Definitions • The minimum genus even embeddings • Cycle parities • Rotation systems and current graphs • Problems and main theorems Cycles in Graphs

18. An embedding of a graph • Give a vertex set and its rotation system . These decide an embedding . Cycles in Graphs

19. Example of a rotationsystem a rotation system an embedding of a graph Cycles in Graphs

20. Embeddings on nonorientable surfaces • Give signs to the edges • We call a twisted arc if . We embed such that each of rotation of and is reverse. Cycles in Graphs

21. Current graphs • A current graph is a weighted embedded directed graph , where . • We call twisted arcs broken arcs. Cycles in Graphs

22. Derived graphs • A current graph derives a derived graph as follows. • Sequences of currents on the face boundaries of become : rotation of . • is defined by adding for each element of . Cycles in Graphs

23. Derived graphs • We define so that arcs which are traced same direction in face boundaries become twisted arcs, and the others become nontwisted arcs. Cycles in Graphs

24. current graph rotationsystem derived graph Cycles in Graphs

25. Outline • Definitions • The minimum genus even embeddings • Cycle parities • Rotation systems and current graphs • Problems and theorems Cycles in Graphs

26. Problem • Which is the type of the cycle parities of the even embeddings of the complete graphs derived from current graphs? Cycles in Graphs

27. Theorem 2 Theorem 2 Let be a current graph which derives : nontrivial even embedding. All arcs are traced same direction in face boundaries of , if and only if is in type A or E. Cycles in Graphs

28. Theorem 3 Theorem 3 Let be a current graphwith m broken arcs with which derives : nontrivial even embedding. Then, the cycle parity is in either typeA, B or F if m is odd, either typeC, D or E if m is even. Cycles in Graphs

29. A list of the cycle parity Cycles in Graphs

30. Type Type AD Type Type B E TypeType C F Cycles in Graphs

31. Future work • The other cases • How is the ratio of embeddings in each type of the cycle parity in all the minimum genus even embeddings of ? Cycles in Graphs

32. Thank you for your attention! Cycles in Graphs

33. Cycles in Graphs