Small cycle cover of 3-connected cubic graphs

1. Small cycle cover of 3-connected cubic graphs 2009.7.29 Fan Yang (杨帆) and Xiangwen Li (李相文) Dep. of Mathematics Huazhong Normal University, Wuhan, China

2. Basic Definition • A cycle cover of a graph is a collection of such that every edge of lies in at least one member of .

3. Basic Definition • A cycle double cover of a graph is a cycle cover of such that each edge of lies in exactly two members of . (a) (b) (c)

4. Background • Cycle double cover conjecture: [Szekeres (B.A.M.S,8,1973, p.367-387) and Seymour (AP,1979, p.342-355)] Every bridgeless graph has a cycle double cover. • Bondy(KAP,1990, p.21-40) conjectured: Every 2-connected simple cubic graph on vertices admits a double cycle cover with .

5. Background • Bondy (KAP,1990, p.21-40) conjectured: If is a 2-connected simple graph with vertices, then the edges of can be covered by at most cycles. • Fan (J.C.T.S.B 84,2002,p.54-83) proved this conjecture (By showing it holds for all simple 2-connected graphs).

6. Background • Lai and Li (DM 269, 2003, 295-302) proved: Every 2-connected simple cubic graph on vertices admits a cycle cover with . What about 3-connected simple cubic graph ?

7. Our Result • Theorem: Let be a 3-connected simple cubic graph of order . has a cycle cover with if and only if . ( ) counter examples

8. Graphs of

9. Proof: Necessary • If , then dose not have a cycle cover with . • Eg. is Petersen graph, we know that it is non-hamiltonian. So it needs at least 3 cycles that cover all its edges.

10. G non-triangle Case 1 Case 2 Case 3 Case 4 Case 5 is a 3-connected simple cubic graph of order . If , then has a cycle cover with . Proof: Sufficiency • Case 1. G contains a triangle • Case 2. G has a minimal nontrivial 3-edge cut. • Case 3. G has a minimal nontrivial 4-edge cut. • Case 4. G has a minimal nontrivial 5-edge cut. • Case 5. G has a minimal nontrivial k-edge cut (k>=6).

11. Nontrivial k-edge cut • Nontrivial k-edge cut: Let be a k-edge cut of . If are pairwise nonadjacent edges of , is called a nontrivial k-edge cut of . Eg.

12. Minimal nontrivial k-edge cut • Minimal nontrivial -edge cut: If is a nontrivial -edge cut of and for any edge cut of with , is not a nontrivial edge cut of , Then is called a minimal nontrivial -edge cut of .

13. Proof: Case 1 Case 1. contains a triangle • If has a cycle cover , then has a cycle cover such that .

14. Proof: Case 2 • Case 2. has a minimal nontrivial 3-edge cut .

15. Proof: Case 2

16. Proof: Case 3 • Case 3. has a minimal nontrivial 4-edge cut . • has a nontrivial 3-cut • If has a cycle cover , then has a cycle cover with . • If has a cycle cover , then has a cycle cover with .

17. Definition: • Removal of an edge Let , Remove and to replace the paths and by the edges and , respectively. • Denote by the resulting graph.

18. Proof: Case 3 • has a cycle cover such that

19. Proof: Case 4 • Case 4. has a minimal nontrivial 5-edge cut . • has a minimal nontrivial 3-edge cut • By induction, has a cycle cover such that

20. Proof: Case 4 • has a cycle cover such that = • has a cycle cover such that

21. Proof: Case 5 • Case 5. has a minimal nontrivial edge cut with . Then graph has a minimal nontrivial edge cut with & • By induction, has a cycle cover with . • has a cycle cover with

22. Our Result • Theorem: Let be a 3-connected simple cubic graph of order . has a cycle cover with if and only if .

23. 谢谢各位老师和同学！ 问题 ?

24. Lemmas • Theorem 1(Lovasz, Roberson). Let be a set of three pairwise-nonadjacent edges in a simple 3-connected graph . Then there is a cycle of containing all three edges of unless is an edge cut of .

25. Results • Lemma 2. Let and be any edge of . If is not an edge of any triangle, then there is a cycle cover of such that .

26. Results • Lemma 4. Suppose that is a graph shown in Fig. 1. For any vertex , let . Then for any given 2-paths and where , has a cycle cover such that contains and contains .

27. Results • Lemma 3. Let , and Then there is a cycle cover of such that contains path , contains path , contains path .

28. Lemmas • Lemma 6. Let be a triangle free simple cubic graph. If is a minimal nontrivial -edge connected graph and , then is a minimal nontrivial -connected simple cubic graph.

29. Lemmas • Lemma 7.

30. Sufficiency • Case 3. has a minimal nontrivial 4-edge cut such that has a component with .

31. Proof of Theorem • By induction, has a cycle cover such that . • Then has a cycle cover such that . • So has a cycle cover such that .

32. Proof of Theorem • Case 4. has a minimal nontrivial 5-edge cut such that has a component with . • contains a triangle. • Contract this triangle, get graph • By induction, has a cycle cover such that

33. Proof of Theorem • has a cycle cover such that • has a cycle cover such that • = • has a cycle cover such that

34. Outline of Proof • Lemma 1. Let be a 3-connected simple cubic graph on vertices. does not have a cycle cover with if and only if is one of and .

35. Proof of Theorem • Proof of Theorem: From Lemma 1, it is sufficient to show that when , has a cycle cover with . • By contraction, • is minimized.

36. Background • Lai and Li (DM 269, 2003, 295-302) proved: Every 2-connected simple cubic graph on vertices admits a cycle cover with . • Barnette(J.C.M.C.C.20,1996, 245-253) proved: If is a 3-connected simple planar graph of order , then the edges of can be covered by at most cycles.