1. Student：連敏筠 Advisor：傅恆霖
Cycle Cover of Graphs
National Chiao Tung University
Student：連敏筠 Advisor：傅恆霖
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2. Outline Introduction and Preliminaries Main Results Conclusion
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3. 1. Introduction and Preliminaries
Definition. (Cycle cover)
A cycle cover of G is a collection C of cycles of G which covers all edges of G; C is called a cycle m-cover if each edge of G is covered exactly m times.
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4. Example: Cycle Cover G： C：
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5. Example: Cycle 2-Cover G: C：
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6. Definition. (Contraction) Contract an edge e.
contract e e
Contract a subgraph F.
contract F F
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7. Definition. (Subdivision) Subdivision an edge xy. x x x H: H: G: zw y y y
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8. Definition. (Isomorphism)
An isomorphism from a graph G to a graph H is a bijection
Definition. (Homeomorphic)
Two graphs G and G' are homeomorphic if there is an isomorphism from some subdivision of G to some subdivision of G'.
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9. Definition. (Integer Flow)
An orientation of an undirected graph is an assignment of a direction to each edge.
For each vertex , v
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10. v
A flow in G under orientation D is an integer-valued function on E(G) such that
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11. The support of is defined by
For a positive k, if then is called a k-flow.
If , then is called a nowhere-zero k-flow.
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12. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Example: nowhere-zero 2-flow
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13. Let G be an r-regular and r-edge-connected graph then G is an r-graph.
Definition. (r-graph)
An r-graph G is an r-regular graph such that for each vertex subset with
Lemma.
Let G be an r-regular and r-edge-connected graph then G is an r-graph.
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14. Theorem 1. (Edmond [1965])
Let G be an r-graph. Then there is an integer p and a family M of perfect matchings such that each edge of G is contained in exactly p members of M.
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15. Definition. (weighted graph)
A graph is called a weighted graph if each edge e is assigned a nonnegative real number , called a weight of e. Let G be a weighted graph and H a subgraph of G. The weight of H is defined by
Theorem 2.
Let G be an r-graph where is odd. Then G contains a
perfect matching M such that
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16. Theorem. (Alon and Tarsi [1985])
Every bridgeless graph G has a cycle cover of
size at most
Conjecture. (Tutte’s 3-flow conjecture)
Every 4-edge-connected graph has a nowhere-zero 3-flow.
Definition： For a flow in G， we define
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17. (2) If G has a nowhere-zero 6-flow such that
Theorem. (Fan [1993])
If Tutte’s 3-flow conjecture is true, then every bridgeless graph G has a nowhere-zero 6-flow such that
(2) If G has a nowhere-zero 6-flow such that
, then G has a cycle cover in which the
sum of lengths of the cycles in the cycle cover is at most
※ bridgeless graph = 2-edge-connected.
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18. 2. Main Results If Tutte’s 3-flow Conjecture is true. (1) (k is odd)
Every graph which is homeomorphic to (k-1)-edge-
connected graph G with has a nowhere-zero 6-
flow such that , then G has a cycle
cover in which the size of the cycles is at most
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19. Every graph which is homeomorphic to (k-1)-edge-
Main Theorem
(2) (k is even)
Every graph which is homeomorphic to (k-1)-edge-
connected graph G with has a nowhere-zero 6-
flow such that , then G has a cycle
cover in which the size of the cycles is at most
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20. If Tutte’s 3-Flow Conjecture is true, then every graph which
Consider k is odd.
Theorem 3.
If Tutte’s 3-Flow Conjecture is true, then every graph which
is homeomorphic to a (k-1)-edge-connected graph G with
has an even subgraph F and 3-flow such that
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21. If Tutte’s 3-Flow Conjecture is true, then every weighted (k-
Theorem 4.
If Tutte’s 3-Flow Conjecture is true, then every weighted (k-
1)-edge-connected graph G with has an even
subgraph F and 3-flow such that
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22. Proof:
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23. Lemma 1.
Let G be an (r-1)-edge-connected graph with Then for each , there exist two edges and such that is (r-1)-edge-connected.
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24. Let G be an (k-1)-edge-connected graph with δ(G) = k.
Fact 1.
Let G be an (k-1)-edge-connected graph with δ(G) = k. y G’ G x z
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25. Lemma 3.
If Tutte’s 3-Flow Conjecture is true, then every weighted (k-1)-edge-connected graph G (k is odd) has an even subgraph F and 3-flow such that
Lemma 4.
If G is a counterexample of Theorem 4 with a minimum number of edges, then G is simple, k-regular and k-edge-connected.
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26. Theorem 5. If a (k-1)-edge-connected graph G with (k is odd)
has a nowhere-zero 6-flow such that
, then G has a cycle cover in which the size of the cycle
cover is at most
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27. 3. Conclusion Theorem. (Fan)
If Tutte’s 3-flow conjecture is true than G has a nowhere-
zero 6-flow such that , then
G has a cycle cover in which the sum of lengths of the
cycles in the cycle cover is at most
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28. If Tutte’s 3-flow conjecture then every graph which is
Main Theorem.
(1) k is odd
If Tutte’s 3-flow conjecture then every graph which is
homeomorphic to a (k-1)-edge-connected graph G with
has a nowhere-zero 6-flow such that
, then G has a cycle cover in
which the size of the cycles is at most
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29. If Tutte’s 3-flow conjecture then every graph which is
(2) k is even
If Tutte’s 3-flow conjecture then every graph which is
homeomorphic to a (k-1)-edge-connected graph G with
has a nowhere-zero 6-flow such that
, then G has a cycle cover in
which the size of the cycles is at most
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30. Fan’s result can be obtained using .
For ,the upper bound we obtained in this paper is
asymptotically approaching to which is much better
than
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31. Thank you for your attention!
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