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

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

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)

Background Cycle double cover conjecture: [Szekeres (B.A.M.S,8,1973, p ) and Seymour (AP,1979, p )] 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.

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).

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

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

Graphs of

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.

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). G Case 1 Case 2 Case 3 Case 4 Case 5 non-triangle is a 3-connected simple cubic graph of order. If, then has a cycle cover with.

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. Nontrivial k-edge cut Eg.

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.

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

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

Proof: Case 2

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.

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

Proof: Case 3 has a cycle cover such that

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

Proof: Case 4 has a cycle cover such that = has a cycle cover such that

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

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

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.

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.

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.

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

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.

Lemmas Lemma 7.

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

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.

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

Proof of Theorem has a cycle cover such that = has a cycle cover such that

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.

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.

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