small cycle cover of 3-connected cubic graphs fan yang ( 杨帆 ) and xiangwen li ( 李相文 ) dep....

Small cycle cover of 3-connected cubic graphs

Fan Yang ( 杨帆 ) and Xiangwen Li ( 李相文 )

Dep. of MathematicsHuazhong Normal University, Wuhan, China

2009.7.29

Basic Definition

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

G G

1 1 2 3 4 4 3 2 1 5 5 1C u u u u v v v v v u u

2 1 2 2 3 3 4 5 5 1 1C u u v v u u u v v u

3 1 2 3 4 5 1C v v v v v v

1 2 3{ , , }C C C

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 .

1C 2C 3C

G G

1 2 3{ , , }C C C

G

(a) (b) (c)

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 .

n

2n

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).

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(2 3) 3n

Background

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

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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 .

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G 6n G 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.

G G

6n

2 10 6 2G G

Proof: Sufficiency

• Case 1. G contains a triangle

• Case 2. G has a minimal nontrivial 3-edge cut.



• Case 5. G has a minimal nontrivial k-edge cut (k>=6).

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Case1 Case

2Case

3Case

4Case

5

non-triangle

G 8nG 6n G

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 .

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G TG

1 2, , , ke e e

1e

2e

3e

4e

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 .

k

T k G*T G *| |T k *T

G

T kG

Proof: Case 1

G *G

Case 1. contains a triangle

G

• If has a cycle cover , then has a cycle cover such that .

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G*G * *| | | |

Proof: Case 2

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

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Proof: Case 2

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'C

Proof: Case 3


• 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 .

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*1( )G e

1 2 3 4{ , , , }T e e e e

'2 3 4{ , , }T e e e

*( )G e

G e

G

G e 1

Definition:

• Removal of an edge Let , Remove and to replace the paths and by the edges and , respectively.

• Denote by the resulting graph.

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1 1x uy

2 2x vy1 1x y

2 2x y

*( )G e

*( )G e

u v

1x 1y 2x 2y

G

e

*| ( ) | | ( ) | 2V G e V G

*1( )G e

u v

1x 1y 2x 2y

Proof: Case 3

• has a cycle cover such that

*1( )G e

G

Proof: Case 4


• has a minimal nontrivial 3-edge cut

• By induction, has a cycle cover such that

G1 2 3 4 5{ , , , , }T e e e e e

( , )i je e T' { , }i jT T e e

( 8) 6n

Proof: Case 4


=

• has a cycle cover such that G

{ , }i jG e e

( 8) 6n

( 8) 6 1 6n n

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

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* *1 2 3( { , , })G G e e e 'T '| | 3T

*| ( ) | 6 8 ( 14)V G n n

*G

G

Our Result

• Theorem: Let be a 3-connected simple cubic graph of order .

has a cycle cover with if and only if .

G8n

G 6n G

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 .

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GS

SG

G

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 .

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1 2 3{ , , }C C C G

1 2 ( )C C E G e

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 .

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1 2 3( ) { , , }N v x x x

1 i jP x vx 2 i kP x vx { , , } {1,2,3}i j k G

1 2{ , }C C 1C

1P 2C 2P

Results

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

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1 2 3{ , , }C C CG 1C 1 2x vx 2C

1 3x vx3C 2 3x vx

v

1x

2x 3x

1C 2C

3C

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.

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( )e E G *( )G e

( 1)k

Lemmas

• Lemma 7.

Sufficiency

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

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1 2 3 4{ , , , }T e e e e

1e

e

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 .

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'G

| ( ) |V G n *1| (( { , }) ) | 4V G e e n

'| ( ) | 6V G n

'G

G

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

GG TT

H ( ) 5V H

1 2 3 4 5{ , , , , }T e e e e e

( , )i je e T1| ( ) | 5V H

'G'G

Proof of Theorem



• =


*( { , })i jG e e

G

{ , }i jG e e

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 .

G8 12n G

6n G 1G 2 3 4 5, , ,G G G G

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.

14n G

6n

( )V G

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.

Gn G

( 1) 2n

6n 4n

small cycle cover of 3-connected cubic graphs fan yang ( 杨帆 ) and xiangwen li ( 李相文 ) dep....

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