Kenta Noguchi Keio University Japan 2012/5/301Cycles in
Graphs
Slide 2
Outline Definitions The minimum genus even embeddings Cycle
parities Rotation systems and current graphs Problems and main
theorems 2012/5/302Cycles in Graphs
Slide 3
Outline Definitions The minimum genus even embeddings Cycle
parities Rotation systems and current graphs Problems and main
theorems 2012/5/303Cycles in Graphs
Slide 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 even embedding : an embedding which has no odd faces
2012/5/304Cycles in Graphs
Slide 5
Outline Definitions The minimum genus even embeddings Cycle
parities Rotation systems and current graphs Problems and main
theorems 2012/5/305Cycles in Graphs
Slide 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.
2012/5/306Cycles in Graphs
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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
Slide 8
A list of the cycle parity TrivialNontrivial TrivialType AType
BType C TrivialType DType EType F 2012/5/308Cycles in Graphs
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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
Slide 10
Outline Definitions The minimum genus embeddings Cycle parities
Rotation systems and current graphs Problems and main theorems
2012/5/3010Cycles in Graphs
Slide 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
2012/5/3011Cycles in Graphs
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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
Slide 13
Equivalence classes on Trivial (bipartite) Nontrivial
(nonbipartite) 2012/5/3013Cycles in Graphs
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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
Slide 15
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
Slide 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. 2012/5/3016Cycles in
Graphs 2012/5/3016Cycles in Graphs
Slide 17
Outline Definitions The minimum genus even embeddings Cycle
parities Rotation systems and current graphs Problems and main
theorems 2012/5/3017Cycles in Graphs
Slide 18
An embedding of a graph Give a vertex set and its rotation
system. These decide an embedding. 2012/5/3018Cycles in Graphs
Slide 19
Example of a rotation system a rotation system an embedding of
a graph 2012/5/3019Cycles in Graphs
Slide 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. 2012/5/3020Cycles in Graphs
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Current graphs A current graph is a weighted embedded directed
graph, where. We call twisted arcs broken arcs. 2012/5/3021Cycles
in Graphs
Slide 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.
2012/5/3022Cycles in Graphs
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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
Slide 24
current graph rotation system derived graph 2012/5/3024Cycles
in Graphs
Slide 25
Outline Definitions The minimum genus even embeddings Cycle
parities Rotation systems and current graphs Problems and theorems
2012/5/3025Cycles in Graphs
Slide 26
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
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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
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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
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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
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Type A D Type B E Type C F 2012/5/3030Cycles in Graphs
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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
Slide 32
Thank you for your attention! 2012/5/3032Cycles in Graphs
Slide 33
Type AType BType C ( ?) ?? ? Type DType EType F ( ?) ? ?
2012/5/3033Cycles in Graphs