the contractible subgraph of k-connected graphs...[t1] w.t. tutte, a theory of 3-connected graphs....

The contractible subgraph of k-connected graphs

Dedicated to Professor Y.Egawa on the occasion of his 60th bithday

Chengfu Qin

Email: [email protected]

Department of mathematics, Guangxi Teachers Education University, Nanning,

Guangxi

2013.9 Tokyo University of Science

Chengfu Qin () The contractible subgraph of...2013.9 Tokyo University of Science 1 /

1 Some related Backgrounds

2 Contraction critical 5-connected graph

Vertex transitive and contraction critical 5-connected graph

Minimal contraction critical 5-connected graph

3 The contractible subgraph of 5-connected graph

4 Minor minimally 5-connected graph

5 Contraction Critical k-connected graph

Some related Backgrounds

Definition 1A subgraph 𝐻 of a k-connected graph is said to be k- contractible if its

contraction, that is, identification every component of 𝐻 to a single vertex, results

still a k-connected graph. When 𝐻 is an edge, we say it is a k-contractible edge.

Definition 2A k-connected graph without k-contractible edge is said to be contraction critical

𝑘-connected graph(briefly, we say it is a k-connected CC-graph).

Definition 1A subgraph 𝐻 of a k-connected graph is said to be k- contractible if its

contraction, that is, identification every component of 𝐻 to a single vertex, results

still a k-connected graph. When 𝐻 is an edge, we say it is a k-contractible edge.

Definition 2A k-connected graph without k-contractible edge is said to be contraction critical

𝑘-connected graph(briefly, we say it is a k-connected CC-graph).

Tutte’s proved that every 3-connected graph on more than four vertices

contains an edge whose contraction yields a new 3-connected graph, [T1].

For 𝑘- connected graphs with 𝑘 ≥ 4, there are infinitely many nonisomorphic

𝑘- connected CC-graphs. However, every 4-connected graph on at least seven

vertices can be reduced to a smaller 4-connected graph by contracting one or two

edges subsequently.

[T1] W.T. Tutte, A theory of 3-connected graphs. Nederl. AKad. Wetensch.

Por. Ser. A 64(1961), 441–455.

Tutte’s proved that every 3-connected graph on more than four vertices

contains an edge whose contraction yields a new 3-connected graph, [T1].

For 𝑘- connected graphs with 𝑘 ≥ 4, there are infinitely many nonisomorphic

𝑘- connected CC-graphs. However, every 4-connected graph on at least seven

vertices can be reduced to a smaller 4-connected graph by contracting one or two

edges subsequently.

[T1] W.T. Tutte, A theory of 3-connected graphs. Nederl. AKad. Wetensch.

Por. Ser. A 64(1961), 441–455.

The existence and the distribution of 𝑘-contractible subgraphs is an

attractive research area within graph connectivity theory. Also it is close relate to

graph minor theorey.

Definition 3For two graphs 𝐺 and 𝐻, we say that 𝐻 is a minor of 𝐺 if we can get 𝐻 from 𝐺

by the following operations: (1) remove edges; (2) delete vertices; (3) contracts

subgraph.

In 2001, Dr.Fijav𝑧 posted the following conjecture.

Conjecture 1

Every 5-connected graph contains one of the graphs 𝐾6, 𝐾2,2,2,1, 𝐶5 + 𝐾3, 𝐼, ̃︀𝐼and 𝐺0 as minor.

There are some results relate to Conjecture 1.

(W. Mader )Every 5-connected planar graph has icosahedron as minor.

( G.Fijav𝑧) Conjecture 1 holds for 5-connected graph with at most 10

vertices and 5-regular 5-connected graph with at most 12 vertices, [GF].

( G.Fijav𝑧) Every 5-connected projective graph with face-width at least three

has 𝐾6 as minor, [GFBM].

[GF] G.Fijav𝑧,Graph minors and connectivity[D], Ljubljana: University of

Ljubljana, 2001.

[GFBM] G.Fijav𝑧, B.Mohar, 𝐾6-minors in projective planar graphs,

Combinatorica, 23(3)(2003): 453-465.

Ljubljana, 2001.

Combinatorica, 23(3)(2003): 453-465.

Ljubljana, 2001.

Combinatorica, 23(3)(2003): 453-465.

Ljubljana, 2001.

Combinatorica, 23(3)(2003): 453-465.

In order to proved Conjecture 1, Professor Kriesell posted the following

conjecture.

Conjecture 2

( M. Kriesell,2007)There are positive integer 𝑏(𝑘) and ℎ(𝑘), such that any

𝑘-connected graph with 𝑏(𝑘) vertices has a 𝑘- contractible subgraph 𝐻 such that

𝐻 has at most ℎ(𝑘) vertices.

For 𝑘 = 2, 3 and 4, one can show that the Conjecture 2 holds. For 𝑘 ≥ 6,

Professor Kriesell gave some examples to show that the conjecture failed.

Further, Professor Kriesell showed the following theorem held.

Theorem 1.1

( M. Kriesell) Every essentially-6-connected graph with at least 13 vertices has a

5- contractible subgraph 𝐻 on at most five vertices, [MK].

[MK] M.Kriesell, How to contract an essentially 6-connected graph to a

5-connected graph, Discrete Mathematics,2007(307):494-510.

For 𝑘 = 2, 3 and 4, one can show that the Conjecture 2 holds. For 𝑘 ≥ 6,

Professor Kriesell gave some examples to show that the conjecture failed.

Further, Professor Kriesell showed the following theorem held.

Theorem 1.1

( M. Kriesell) Every essentially-6-connected graph with at least 13 vertices has a

5- contractible subgraph 𝐻 on at most five vertices, [MK].

Vertex transitive and contraction critical 5-connected graph

Minimal contraction critical 5-connected graph

Properties of contraction critical 5-connected graph

Definition 4Let 𝐺 be 𝑘-connected graph, 𝑇 be a set of 𝑘 vertices. If 𝐺− 𝑇 has at least two

components, then we say that 𝑇 is a separator of 𝐺. The union of at least one

but not of all the components of 𝐺− 𝑇 is called a fragment.

In 1991, Professor Egawa proved the following result.

Theorem 2.1Let 𝑘 be an integer greater than two, and G be a 𝑘−connected graph not

isomorphic to 𝐾𝑘+1. If G is CC-graph, then 𝐺 has a fragment of cardinality at

most 𝑘4

Thus, for 𝑘 ≤ 7, every 𝑘-connected CC-graph contains at least one vertex of

degree 𝑘.

most 𝑘4

degree 𝑘.

most 𝑘4

degree 𝑘.

Theorem 2.2

(Su Jianji, Kiyoshi. Ando, independently ) Every vertex of 5-connected CC-graph

adjacent to at least two vertices of degree 5 and, hence, there at least 25 |𝐺|

vertices of degree 5 .

[Su ]Su Jianji, Vertices of degree 5 in contraction critical 5-connected graphs.

J. Guangxi Normal University, 3,12-16 (1997)(in chinese)

[Ando ] Kiyoshi. Ando, A Local Structure Theorem on 5-Connected Graphs,

J Graph Theory, 60, 99–129(2009)

Theorem 2.2

(Su Jianji, Kiyoshi. Ando, independently ) Every vertex of 5-connected CC-graph

adjacent to at least two vertices of degree 5 and, hence, there at least 25 |𝐺|

vertices of degree 5 .

[Su ]Su Jianji, Vertices of degree 5 in contraction critical 5-connected graphs.

J. Guangxi Normal University, 3,12-16 (1997)(in chinese)

[Ando ] Kiyoshi. Ando, A Local Structure Theorem on 5-Connected Graphs,

J Graph Theory, 60, 99–129(2009)

Problem 1Determine the largest constant c such that every 5-connected CC-graph of order

n has at least 𝑐𝑛 vertices of degree 5.

Clearly, 𝑐 ≥ 25 . Professor Ando gave an example to show that 𝑐 ≤ 8

Contraction critical 5-connected graph

We study the subgraph of a 5-connected CC-graph which is induced by the

set of vertices of degree five. We have the following properties of 5-connected

CC-graphs.

Lemma 1

Let 𝐺 be a 5-connected CC-graph and let 𝐻 = 𝐺[𝑉5(𝐺)], 𝐻0 be a component of

𝐻, then the following holds.

(1) ∆(𝐻0) ≥ 3 has at least four vertices.

(2) 𝐻0 has at least four vertices. Further, if 𝐻0 has exactly four vertices, then

𝐻0∼= 𝐾−

Professor Ando characterized the local structure of component of 𝐻 which

has exactly four vertices.

(a) (b)

Lemma 2Let 𝐺 be a 5-connected CC-graph , there exist a vertex 𝑥 of 𝐺 such that every

edge that incident with 𝑥 is contained in some triangles.

Lemma 3Let 𝐺 be a 5-connected CC-graph. Let 𝑥 be a vertex of 𝐺 and

{𝑥1, 𝑥2} ⊆ 𝑉5(𝐺) ∩𝑁(𝑥). If 𝑥1𝑥2 ∈ 𝐸(𝐺) and 𝑑(𝑥) ≥ 8, then

|𝑉5(𝐺) ∩𝑁(𝑥)| ≥ 3.

In 2010, Li Tingting showed that the condition 𝑑(𝑥) ≥ 8 can be reduced to

𝑑(𝑥) ≥ 6.

Lemma 2Let 𝐺 be a 5-connected CC-graph , there exist a vertex 𝑥 of 𝐺 such that every

edge that incident with 𝑥 is contained in some triangles.

Lemma 3Let 𝐺 be a 5-connected CC-graph. Let 𝑥 be a vertex of 𝐺 and

{𝑥1, 𝑥2} ⊆ 𝑉5(𝐺) ∩𝑁(𝑥). If 𝑥1𝑥2 ∈ 𝐸(𝐺) and 𝑑(𝑥) ≥ 8, then

|𝑉5(𝐺) ∩𝑁(𝑥)| ≥ 3.

In 2010, Li Tingting showed that the condition 𝑑(𝑥) ≥ 8 can be reduced to

𝑑(𝑥) ≥ 6.

By Lemma 1, Lemma 2 and Lemma 3, we have the following Theorem

Theorem 2.3

Let 𝐺 be a 5-connected CC-graph, then 𝑐 ≥ 49 (|𝑉5(𝐺)| ≥ 4

9 |𝐺|).

Later, Li Tingting et.al and Professor Ando et.al, independently, showed that

𝑐 ≥ 12 (|𝑉5(𝐺)| ≥ 1

2 |𝐺|). Further, Professor Ando gave an example to showed that

this result is best possible.

[Li ] Li Tingting, Su Jianji, A new lower bound of the munber of degree 5 in

5-connected CC-graph, Graphs and Combinatorics, 26(3)(2010):395-406.

[Ando ] K. Ando and T. Iwase, The number of vertices of degree 5 in a

contractioncritically 5-connected graph, Discrete Math 331 (2011),

1925–1939.

Theorem 2.3

9 |𝐺|).

𝑐 ≥ 12 (|𝑉5(𝐺)| ≥ 1

1925–1939.

Theorem 2.3

9 |𝐺|).

𝑐 ≥ 12 (|𝑉5(𝐺)| ≥ 1

1925–1939.

Vertex transitive and contraction critical 5-connected

Lemma 4

Let 𝑝 be a prime integer and 𝐺 be a vertex transitive graph with 𝜅(𝐺) = 𝑝, then

𝐺 is 𝑝 regular.

Lemma 5Let 𝐺 be a vertex transitive 5-connected CC-graph, then 𝐺 is 5-regular and every

edge of 𝐺 is contained in triangle.

Lemma 4

Let 𝑝 be a prime integer and 𝐺 be a vertex transitive graph with 𝜅(𝐺) = 𝑝, then

𝐺 is 𝑝 regular.

Lemma 5Let 𝐺 be a vertex transitive 5-connected CC-graph, then 𝐺 is 5-regular and every

edge of 𝐺 is contained in triangle.

Theorem 2.4

Let 𝐺 be a vertex transitive 5-connected CC-graph. If |𝑉 (𝐺)| ≤ 9, then either

𝐺 ∼= 𝐾6 or 𝐺 ∼= 𝐺*.

(c) 𝐺*

Theorem 2.5

Let 𝐺 be a 5- connected CC-graph with |𝑉 (𝐺)| ≥ 10. If 𝐺 is vertex transitive

graph, then for any 𝑥 ∈ 𝑉 (𝐺), 𝑥 has type 1, 2, 3 or 4.

(d) Type 1 (e) Type 2

(f) Type 3 (g) Type 4

Theorem 2.6

Let 𝐺 be a vertex transitive 5-connected CC-graph with |𝑉 (𝐺)| ≥ 10, let 𝑥 be a

vertex of 𝐺. If 𝑥 has type 2, then 𝐺 is icosahedron.

Minimally 5-connected graph

Definition 5Let 𝐺 be a 𝑘-connected graph, if for any edge 𝑒 of 𝐺, 𝐺− 𝑒 is not a

𝑘-connected, then we say that 𝐺 is minimally 𝑘-connected graph.

By definition, we know that for any edge 𝑥𝑦 of minimally 𝑘-connected graph,

there are a vertex set 𝑇 in 𝐺− 𝑥𝑦 which separates 𝑥 and 𝑦 and |𝑇 | = 𝑘 − 1.

minimally contraction critical 5-connected graph

Professor Ando and C. Qin studied the structure of minimally 5-connected

CC-graph.

Theorem 2.7

(Kiyoshi.Ando and Chengfu Qin) Let 𝐺 be a minimally 5-connected CC-graph, if

𝐺[𝑉≥6] has edge 𝑥𝑦, then there are one of the following two configuration around

𝑥𝑦. Here [𝑉≥6] denote the set of vertices with degree 6.

The contractible subgraph of 5-connected graph

Theorem 3.1

Let 𝐺 be 5- connected graph such that 𝐺[𝑉≥6] has some edges, then 𝐺 has a

contractible subgraph with cardinality at most four or 𝐺 has removable edge.

Theorem 3.2Let 𝐺 be a 5-connected graph and 𝐺 has a vertex, say 𝑥, which dose not

contained in any triangle, then 𝐺 has a contractible subgraph with cardinality at

most four.

Theorem 3.3Let 𝐺 be a 5-connected graph other than icosahedron. If 𝐺 is vertex transitive,

then 𝐺 has a contractible subgraph with cardinality at most four.

(m) (n) (o)

Minor minimally 5-connected graph

Definition 6For two graphs 𝐺 and 𝐻, we say that 𝐻 is a minor of 𝐺 if we can get 𝐻 from 𝐺

by the following operations: (1) remove edges; (2) delete vertices; (3) contracts

subgraph.

Definition 7Let 𝐺 be a 5-connected graph and 𝐺 dose not contain an other 5-connected

graph as minor, then we say that 𝐺 is Minor minimal 5-connected graph.

Corollary 4.1

Let 𝐺 be a minor minimally 5-connected graph, then every edge of 𝐺 incident to

at least one vertex of degree 5.

Corollary 4.2

Let 𝐺 be a minor minimally 5-connected graph with |𝑉 (𝐺)| ≥ 10 , then every

vertex of 𝐺 is contained in some triangles.

Corollary 4.1

Let 𝐺 be a minor minimally 5-connected graph, then every edge of 𝐺 incident to

at least one vertex of degree 5.

Corollary 4.2

Let 𝐺 be a minor minimally 5-connected graph with |𝑉 (𝐺)| ≥ 10 , then every

vertex of 𝐺 is contained in some triangles.

(M.Kriesell) Every minor minimally hyper-5-connected graph has at most 12

vertices, [MK].

Theorem 4.3Let 𝐺 be a minor minimally super-5-connected graph, then 𝐺 isomorphic to one

of 𝐾6,𝐾2,2,2,1, 𝐶5 + 𝐾3, 𝐼 and 𝐼.

Theorem 4.4Let 𝐺 be a minor minimally vertex transitive 5-connected graph, then 𝐺

insomphics to either 𝐾6 or 𝐼.

vertices, [MK].

of 𝐾6,𝐾2,2,2,1, 𝐶5 + 𝐾3, 𝐼 and 𝐼.

vertices, [MK].

of 𝐾6,𝐾2,2,2,1, 𝐶5 + 𝐾3, 𝐼 and 𝐼.

Contraction Critical k-connected graph

Theorem 5.1

(Zhao Qiaofeng, Qin Chengfu ) Let 𝐺 be a 6-connected CC-graph, then

|𝑉6(𝐺)| ≥ 15 |𝑉 (𝐺)|.

Theorem 5.2

(Li Min et.al) Let 𝐺 be a 7-connected CC-graph, then |𝑉7(𝐺)| ≥ 122 |𝑉 (𝐺)|.

Theorem 5.3

Let 𝐺 be a 8-connected CC-graph with 𝛿(𝐺) = 9, then |𝑉9(𝐺)| ≥ 29 |𝑉 (𝐺)|.

Theorem 5.1

|𝑉6(𝐺)| ≥ 15 |𝑉 (𝐺)|.

Theorem 5.2

Theorem 5.3

Theorem 5.1

|𝑉6(𝐺)| ≥ 15 |𝑉 (𝐺)|.

Theorem 5.2

Theorem 5.3

Some Further problems

How to characterize all vertex transitive 5-connected CC-graph?

Dose that 8-connected CC-graph with minimum degree 9 is 9-regular?

Thank You！

the contractible subgraph of k-connected graphs...[t1] w.t. tutte, a theory of 3-connected graphs....

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