Kenta Noguchi Keio University Japan 2012/5/301Cycles in Graphs

Outline Definitions The minimum genus even embeddings Cycle parities Rotation systems and current graphs Problems and main theorems 2012/5/302Cycles in Graphs

Outline Definitions The minimum genus even embeddings Cycle parities Rotation systems and current graphs Problems and main theorems 2012/5/303Cycles in Graphs

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 even embedding : an embedding which has no odd faces 2012/5/304Cycles in Graphs

Outline Definitions The minimum genus even embeddings Cycle parities Rotation systems and current graphs Problems and main theorems 2012/5/305Cycles in Graphs

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. 2012/5/306Cycles in Graphs

Cycle parities In even embedded graphs, the parities of the lengths of homotopic cycles are the same. Then, even embedded graphs can be classified into several types by parities of lengths of their cycles. 2012/5/307Cycles in Graphs

A list of the cycle parity TrivialNontrivial TrivialType AType BType C TrivialType DType EType F 2012/5/308Cycles in Graphs

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. 2012/5/309Cycles in Graphs 2012/5/309Cycles in Graphs

Outline Definitions The minimum genus embeddings Cycle parities Rotation systems and current graphs Problems and main theorems 2012/5/3010Cycles in Graphs

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 2012/5/3011Cycles in Graphs

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. 2012/5/3012Cycles in Graphs

Equivalence classes on Trivial (bipartite) Nontrivial (nonbipartite) 2012/5/3013Cycles in Graphs

Equivalence classes on Trivial Type A Type B Type C 2012/5/3014Cycles in Graphs Theorem (Nakamoto, Negami, Ota) G : locally bipartite graph if G is in type A if G is in type B if G is in type C

Equivalence classes on Trivial Type D Type E Type F 2012/5/3015Cycles in Graphs 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

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. 2012/5/3016Cycles in Graphs 2012/5/3016Cycles in Graphs

Outline Definitions The minimum genus even embeddings Cycle parities Rotation systems and current graphs Problems and main theorems 2012/5/3017Cycles in Graphs

An embedding of a graph Give a vertex set and its rotation system. These decide an embedding. 2012/5/3018Cycles in Graphs

Example of a rotation system a rotation system an embedding of a graph 2012/5/3019Cycles in Graphs

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. 2012/5/3020Cycles in Graphs

Current graphs A current graph is a weighted embedded directed graph, where. We call twisted arcs broken arcs. 2012/5/3021Cycles in Graphs

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. 2012/5/3022Cycles in Graphs

Derived graphs We define so that arcs which are traced same direction in face boundaries become twisted arcs, and the others become nontwisted arcs. 2012/5/3023Cycles in Graphs

current graph rotation system derived graph 2012/5/3024Cycles in Graphs

Outline Definitions The minimum genus even embeddings Cycle parities Rotation systems and current graphs Problems and theorems 2012/5/3025Cycles in Graphs

Problem Which is the type of the cycle parities of the even embeddings of the complete graphs derived from current graphs? 2012/5/3026Cycles in Graphs

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. 2012/5/30Cycles in Graphs27

Theorem 3 Let be a current graph with m broken arcs with which derives : nontrivial even embedding. Then, the cycle parity is in either type A, B or F if m is odd, either type C, D or E if m is even. 2012/5/3028Cycles in Graphs

A list of the cycle parity Theorem 2Theorem 3 Type A ○ Type B ○ Type C ○ (except ) Type E ○ Type F ○ Type D ○ (except ) 2012/5/3029Cycles in Graphs

Type A D Type B E Type C F 2012/5/3030Cycles in Graphs

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 ? 2012/5/3031Cycles in Graphs

Thank you for your attention! 2012/5/3032Cycles in Graphs

Type AType BType C ○○○ ○○ ○ ( ?) ??○ ?○○ Type DType EType F ○○○ ○ ( ?)○○ ○○? ○?○ 2012/5/3033Cycles in Graphs