Student：連敏筠 Advisor：傅恆霖 Cycle Cover of Graphs National Chiao Tung University Student：連敏筠 Advisor：傅恆霖 2019/2/25

Outline Introduction and Preliminaries Main Results Conclusion 2019/2/25

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. 2019/2/25

Example: Cycle Cover G： C： 2019/2/25

Example: Cycle 2-Cover G: C： 2019/2/25

Definition. (Contraction) Contract an edge e. contract e e Contract a subgraph F. contract F F 2019/2/25

Definition. (Subdivision) Subdivision an edge xy. x x x H: H: G: z w y y y 2019/2/25

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'. 2019/2/25

Definition. (Integer Flow) An orientation of an undirected graph is an assignment of a direction to each edge. For each vertex , v 2019/2/25

v A flow in G under orientation D is an integer-valued function on E(G) such that 2019/2/25

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. 2019/2/25

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Example: nowhere-zero 2-flow 2019/2/25

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. 2019/2/25

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. 2019/2/25

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 2019/2/25

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 2019/2/25

(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. 2019/2/25

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 2019/2/25

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 2019/2/25

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 2019/2/25

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 2019/2/25

Proof: 2019/2/25

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. 2019/2/25

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 2019/2/25

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. 2019/2/25

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 2019/2/25

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 2019/2/25

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 2019/2/25

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 2019/2/25

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 . 2019/2/25

Thank you for your attention! 2019/2/25