Extension of Fouquet-Jolivet’s · PDF fileNotations and Definitions Main Results Sketch...

Notations and DefinitionsMain Results

Sketch of Proofs

Extension of Fouquet-Jolivet’s Conjecture

Zhiquan Hu

Faculty of Math. and Stat.

Central China Normal University

Wuhan 430079, PRC

Joint work with

Guantao Chen and Yaping Wu

July 10, 2010, GTCA10

Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture

Sketch of Proofs

Dedicated to Professor Tian’s 70th birthday

Sketch of ProofsNotations and Definitions

Notations and Definitions

I 𝛿(G ): the minimum degree of G

I 𝜎2(G ): min {d(u) + d(v) : u = v , uv /∈ E (G )}

I 𝜇(G ): min {max{d(u), d(v)} : distG (u, v) = 2}

I 𝜅(G ): the connectivity of G

I 𝛼(G ): the independence number of G

I c(G ): the circumference of G

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Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph

Classic results for hamiltonian graphs

I Theorem A (Dirac,1952) Let G be a graph of order n ≥ 3. If

𝛿(G ) ≥ n/2, then G is hamiltonian.

I Theorem B (Ore,1960) Let G be a graph of order n ≥ 3. If 𝜎2 ≥ n,

then G is hamiltonian.

I Theorem C (Chvatal & Erdos,1972) Let G be a k-connected graph

of order n ≥ 3 and 𝛼 be the independence number of G . If 𝛼 ≤ 𝜅

I Theorem D (Fan,1984) Let G be a k-connected graph. If

𝜇(G ) ≥ n/2, then G is hamiltonian.

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Long Cycles Involving Independence Numbers

I Theorem 1 (Chvatal & Erdos,1972) Let G be a k-connected graph

I Conjecture 2 (Fouquet & Jolivet, 1978). Let G be a k-connected

graph of order n. If 𝛼 ≥ k ≥ 2, then c(G ) ≥ k(n+𝛼−k)𝛼 .

I Progress of Fouquet-Jolivet’s Conjecture:

� True for k = 𝛼− 1, 𝛼− 2 (Fournier, 1982)

� True for k = 2 (Fournier, 1984)

� True for k = 3 (Manoussakis, Graphs and Combinators 2009)

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Main ResultsI Theorem 3 (Chen, Hu & Wu, 2008) Let G be a 4-connected graph

of order n with independence number 𝛼. If 𝛼 ≥ 4. Then,

c(G ) ≥ 4(n+𝛼−4)𝛼 .

� The conjecture of of Fouquet and Jolivet is true for k = 4

I Question (asked by a referee of JGT)

� Whether it is possible to prove a weaker result like

c(G ) ≥ k(n+𝛼−k)𝛼 − ck

for a constant ck?

I Theorem 4 (Chen, Hu & Wu, 2009) Let G be a k-connected graph

of order n and independence number 𝛼. If 𝛼 ≥ k ≥ 4, then

c(G ) ≥ k(n+𝛼−k)𝛼 − (k−3)(k−4)

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Main ResultsI Theorem 3 (Chen, Hu & Wu, 2008) Let G be a 4-connected graph

of order n with independence number 𝛼. If 𝛼 ≥ 4. Then,

c(G ) ≥ 4(n+𝛼−4)𝛼 .

� The conjecture of of Fouquet and Jolivet is true for k = 4

I Question (asked by a referee of JGT)

c(G ) ≥ k(n+𝛼−k)𝛼 − ck

for a constant ck?

c(G ) ≥ k(n+𝛼−k)𝛼 − (k−3)(k−4)

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I Theorem 5 (Chen, Hu & Wu, 2009) Let G be a k-connected graph,

k ≥ 2, of order n and independence number 𝛼. If k ≤ 𝛼 ≤ k + 3,

then c(G ) ≥ k(n+𝛼−k)𝛼 .

� The conjecture of of Fouquet and Jolivet is true for k = 𝛼− 3.

I In order to prove Theorems 4 and 5, we proved a key lemma on how

to inserting vertices into a cycle and proposed a conjecture on the

structure of graphs with given independent number 𝛼 (Conjecture

I We proved Conjecture 17 on March 2010 and get the following two

results:

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results:

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results:

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I Theorem 6 (Chen, Hu & Wu, March 2010) Let G be a k-connected

graph with k ≥ 2, let C be a cycle of G and let H be any induced

subgraph of G − V (C ). Then for any real number s ≥ 1,

c(G ) ≥ min{ks, |C |+ |H| − 𝛼(H)(s − 1)},

graph, k ≥ 2, of order n and independence number 𝛼. If 𝛼 ≥ k, then

c(G ) ≥ k(n + 𝛼− k)

𝛼− (p − 1)(k − k0)

where p = ⌊𝛼 ∖ k⌋ and k0 is an integer such that Conjecture 8 is

true for k = k0.

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graph with k ≥ 2, let C be a cycle of G and let H be any induced

subgraph of G − V (C ). Then for any real number s ≥ 1,

c(G ) ≥ min{ks, |C |+ |H| − 𝛼(H)(s − 1)},

graph, k ≥ 2, of order n and independence number 𝛼. If 𝛼 ≥ k, then

c(G ) ≥ k(n + 𝛼− k)

𝛼− (p − 1)(k − k0)

where p = ⌊𝛼 ∖ k⌋ and k0 is an integer such that Conjecture 8 is

true for k = k0.

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I Conjecture 8 (J. Chen, L. Chen, and D. Liu) Let G be a

k-connected graph and k ≥ 2. Then, for any two cycles C1 and C2

in G , there exist two cycles C*1 and C*

2 such that

V (C*1 ) ∪ V (C*

2 ) ⊇ V (C1) ∪ V (C2) and |V (C*1 ) ∩ V (C*

2 )| ≥ k.

I By Theorem 7, the conjecture of of Fouquet and Jolivet is true for

k ≤ 𝛼 ≤ 2k − 1.

I On April 2010, D. B. West told G. Chen that they has just proved

our Conjecture 13 and the full conjecture of of Fouquet and Jolivet.

I Theorem 10 (Suil, West & Wu, 2010) Let G be a k-connected

graph of order n with independence number 𝛼. If 𝛼 ≥ k ≥ 2. Then,

c(G ) ≥ k(n + 𝛼− k)

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2 such that

V (C*1 ) ∪ V (C*

2 ) ⊇ V (C1) ∪ V (C2) and |V (C*1 ) ∩ V (C*

2 )| ≥ k.

k ≤ 𝛼 ≤ 2k − 1.

c(G ) ≥ k(n + 𝛼− k)

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2 such that

V (C*1 ) ∪ V (C*

2 ) ⊇ V (C1) ∪ V (C2) and |V (C*1 ) ∩ V (C*

2 )| ≥ k.

k ≤ 𝛼 ≤ 2k − 1.

c(G ) ≥ k(n + 𝛼− k)

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2 such that

V (C*1 ) ∪ V (C*

2 ) ⊇ V (C1) ∪ V (C2) and |V (C*1 ) ∩ V (C*

2 )| ≥ k.

k ≤ 𝛼 ≤ 2k − 1.

c(G ) ≥ k(n + 𝛼− k)

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I By use Kouider’s Theorem one times and Theorem 6, We proved

the following generalization of Theorem 10.

with k ≥ 2 and let H be a nonempty subgraph of G . Then

c(G ) ≥ min

{|H|, k(|H|+ 𝛼(H)− k)

𝛼(H)

I Theorem (Kouider, 1994) Let G be a k-connected graph, k ≥ 2, of

order n and H be an induced subgraph of G with independence

number 𝛼(H). Then, either the vertices of H are covered by one

cycle of G or else G has a cycle C satisfying

𝛼(H − V (C )) ≤ 𝛼(H)− k.

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c(G ) ≥ min

{|H|, k(|H|+ 𝛼(H)− k)

𝛼(H)

𝛼(H − V (C )) ≤ 𝛼(H)− k.

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c(G ) ≥ min

{|H|, k(|H|+ 𝛼(H)− k)

𝛼(H)

𝛼(H − V (C )) ≤ 𝛼(H)− k .

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c(G ) ≥ k(n + 𝛼− k)

with k ≥ 2 and let H be a nonempty subgraph of G . Then,

c(G ) ≥ min

{|H|, k(|H|+ 𝛼(H)− k)

𝛼(H)

c(G ) ≥ min

{n,max

{k(n + 𝛼− k)

𝛼, k

⌊n + 2𝛼− 2k

⌋}}.

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c(G ) ≥ k(n + 𝛼− k)

c(G ) ≥ min

{|H|, k(|H|+ 𝛼(H)− k)

𝛼(H)

c(G ) ≥ min

{n,max

{k(n + 𝛼− k)

𝛼, k

⌊n + 2𝛼− 2k

⌋}}.

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c(G ) ≥ k(n + 𝛼− k)

c(G ) ≥ min

{|H|, k(|H|+ 𝛼(H)− k)

𝛼(H)

c(G ) ≥ min

{n,max

{k(n + 𝛼− k)

𝛼, k

⌊n + 2𝛼− 2k

⌋}}.

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I Example. Let G := Kk +(kKp ∪mKp−1), where k, p ≥ 2 and m ≥ 1.

� n = k(p + 1) +m(p − 1), 𝜅 = k , and 𝛼 = k +m.

� c(G ) = k(p + 1) = k⌊ n+2𝛼−2k𝛼 ⌋.

� c(G )− k(n+𝛼−k)𝛼 = k[(p+1)− k(p+1)+mp

k+m ] = kmk+m → k (m → ∞).

Therefore, the low bound in Theorem 12 is sharp and better than

that of Theorem 10.

Figure: c(G ) > k⌊n+2𝛼−2k

⌋> k(n+𝛼−k)

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c(G ) ≥ min

{n,max

{k(n + 𝛼− k)

𝛼, k

⌊n + 2𝛼− 2k

⌋}}.

I Remark 1: The function

f (G ) := max{k(|G |+ 𝛼(G )− k)

𝛼(G ), k

⌊|G |+ 2𝛼(G )− 2k

𝛼(G )

from the set of graphs to positive real numbers is not monotonic

increasing according to graph inclusion relation. that is, there exist

a graph G and a subgraph H of G such that f (G ) < f (H).

I Remark 2: If ∅ = H ⊆ G , then f (G [V (H)]) ≥ f (H).

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c(G ) ≥ min

{n,max

{k(n + 𝛼− k)

𝛼, k

⌊n + 2𝛼− 2k

⌋}}.

f (G ) := max{k(|G |+ 𝛼(G )− k)

𝛼(G ), k

⌊|G |+ 2𝛼(G )− 2k

𝛼(G )

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c(G ) ≥ min

{n,max

{k(n + 𝛼− k)

𝛼, k

⌊n + 2𝛼− 2k

⌋}}.

f (G ) := max{k(|G |+ 𝛼(G )− k)

𝛼(G ), k

⌊|G |+ 2𝛼(G )− 2k

𝛼(G )

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I The following is a common generalization of Theorems 10-12.

with k ≥ 2. Then

c(G ) ≥ max { min {|H|, f (H)} : ∅ = H ⊆ G},

where f (H) := max{

k(|H|+𝛼(H)−k)𝛼(H) , k

⌊|H|+2𝛼(H)−2k

𝛼(H)

⌋}I Example. c(G ) = k(p + 1) = min{|H|, f (H)} > f (G ).

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with k ≥ 2. Then

c(G ) ≥ max { min {|H|, f (H)} : ∅ = H ⊆ G},

where f (H) := max{

k(|H|+𝛼(H)−k)𝛼(H) , k

⌊|H|+2𝛼(H)−2k

𝛼(H)

I Example. c(G ) = k(p + 1) = min{|H|, f (H)} > f (G ).

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with k ≥ 2. Then

c(G ) ≥ max { min {|H|, f (H)} : ∅ = H ⊆ G},

where f (H) := max{

k(|H|+𝛼(H)−k)𝛼(H) , k

⌊|H|+2𝛼(H)−2k

𝛼(H)

⌋}I Example. c(G ) = k(p + 1) = min{|H|, f (H)} > f (G ).

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Long Cycles Intersecting a given subgraph

I Let G be a k-connected graph and let V0 a given subset of V (G ). It

is interesting to know that whether V0 is cycliable in G and if V0 is

not cycliable in G , how many vertices of V0 can be contained in one

common cycle of G .

I Theorem 15 (Shi, 1992) Let G be a graph on n vertices and let

W ⊆ V (G ) such that each pair of nonadjacent vertices u, v ∈ W

satisfies d(u) + d(v) ≥ n. If |W | ≥ 3, then G contains a cycle

through all vertices of W .

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Long Cycles Intersecting a given subgraph

I Let G be a k-connected graph and let V0 a given subset of V (G ). It

is interesting to know that whether V0 is cycliable in G and if V0 is

not cycliable in G , how many vertices of V0 can be contained in one

common cycle of G .

I Theorem 15 (Shi, 1992) Let G be a graph on n vertices and let

W ⊆ V (G ) such that each pair of nonadjacent vertices u, v ∈ W

satisfies d(u) + d(v) ≥ n. If |W | ≥ 3, then G contains a cycle

through all vertices of W .

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I We get the following theorem.

with k ≥ 2 and let H be a subgraph of G . If |H| ≥ 𝛼(H) + k, then

there is a cycle C in G such that

|V (C ) ∩ V (H)| ≥ min

{|H|, k

⌊|H|+ 𝛼(H)− k

𝛼(H)

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I We get the following theorem.

with k ≥ 2 and let H be a subgraph of G . If |H| ≥ 𝛼(H) + k, then

there is a cycle C in G such that

|V (C ) ∩ V (H)| ≥ min

{|H|, k

⌊|H|+ 𝛼(H)− k

𝛼(H)

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I Example. Let G := Gk + (mKp ∪ tKp−1), where k, p ≥ 2, t ≥ 1 and

m ≥ k + 1.

For ℓ ≤ t, H := mKp ∪ ℓKp−1 is a subgraph of G with

k×⌊|H|+ 𝛼(H)− k

𝛼(H)

⌋= k×

⌊mp + ℓ(p − 1) + (m + ℓ)− k

m + ℓ

⌋= kp,

which is the maximum number of vertices of H that can be

contained in a common cycle of G . Therefore, the low bound in

Theorem 16 is sharp.

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Ideas of ProofsInserting H-vertices into a cycle C of G − V (H).Structures of Graphs with Given Independence NumberA low bound of c(G) w.r.t. a cycle C and H ⊆ G − C .

Ideas of Proofs

I 1∘ Establish some lemmas on inserting H-vertices into a cycle C of

G − V (H).

2∘ Study the structure of graphs with given independence number.

3∘ Establish a low bound of c(G ) relative to a cycle C and an

induced subgraph H of G − C .

4∘ By using 3∘ and Kouider’s Theorem to get the desired bound.

I Theorem (Kouider, JCTB 1994) Let G be a k-connected graph,

k ≥ 2, of order n and H be an induced subgraph of G . Then, either

the vertices of H are covered by one cycle of G or else G has a cycle

C satisfying 𝛼(H − V (C )) ≤ 𝛼(H)− k.

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Ideas of Proofs

I 1∘ Establish some lemmas on inserting H-vertices into a cycle C of

G − V (H).

2∘ Study the structure of graphs with given independence number.

3∘ Establish a low bound of c(G ) relative to a cycle C and an

induced subgraph H of G − C .

4∘ By using 3∘ and Kouider’s Theorem to get the desired bound.

I Theorem (Kouider, JCTB 1994) Let G be a k-connected graph,

k ≥ 2, of order n and H be an induced subgraph of G . Then, either

the vertices of H are covered by one cycle of G or else G has a cycle

C satisfying 𝛼(H − V (C )) ≤ 𝛼(H)− k .

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Inserting H-vertices into a cycle C of G − V (H)

I Definition Let C be a cycle of G and let H be an induced subgraph

of G − V (C ). For x1 = x2 ∈ V (C ), the segment C [x1, x2] is called a

normal H-interval of C if there exist two internally vertex disjoint

paths P1,P2 in G from V (H) to V (C ) such that

(N-1) V (Pi ) ∩ V (C ) = {xi}, for each i = 1, 2 and

(N-2) |V (H) ∩ (V (P1) ∪ V (P2))| = min {|H|, 2}.

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I Lemma (Hu, Tian & Wei, JCT B82 (2001)) Let m ≥ 0 and k ≥ 2.

Let G be a (m + k)-connected graph and M an m-matching of G .

Let S ⊆ V (G )− V (M) with |S | ≤ k − 2 and let C be a longest

cycle passing through M ∪ S . If l(C ) < min {|V (G )|, 2L−m},where L is a constant, then every component H of G − V (C ) has a

vertex x with dG (x) < L.

I The essential part of the proof: If the Lemma is not true, then there

exists a set of (m + k) pairwise edge disjoint normal H-intervals of

I Question: What happens if m = 0 and H is an induced subgraph of

G − V (C )?

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G − V (C )?

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G − V (C )?

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Inserting vertices into a cycle —a key lemma

I Definition Let V0 ⊆ V (G ). A cycle C of G is called a maximal

V0-cycle if there is no cycle C ′ in G such that V (C )∩V0 is a proper

subset of V (C ′) ∩ V0.

I Key Lemma. Let G be a k-connected graph, k ≥ 2 and s ≥ 1 be

two integers. Let V0 be a subset of V (G ) , let C be a maximal

V0-cycle in G with length at least k and let H be a subgraph of

G [V0 − V (C )] with |H| ≥ s. If every normal H-interval C [x1, x2] of

C satisfies |C (x1, x2) ∩ V0| ≥ s, then

(i) |V (C ) ∩ V0| ≥ ks, and

(ii) |V (C )| ≥ k(s + 1).

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Inserting vertices into a cycle —a key lemma

I Definition Let V0 ⊆ V (G ). A cycle C of G is called a maximal

V0-cycle if there is no cycle C ′ in G such that V (C )∩V0 is a proper

subset of V (C ′) ∩ V0.

G [V0 − V (C )] with |H| ≥ s. If every normal H-interval C [x1, x2] of

C satisfies |C (x1, x2) ∩ V0| ≥ s, then

(i) |V (C ) ∩ V0| ≥ ks, and

(ii) |V (C )| ≥ k(s + 1).

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I How to use the Key Lemma?

By taking V0 = V (G ) in the Key Lemma, we see that if C is a

maximal cycle in G with |C | < k(s + 1), then for every induced

subgraph H of G − V (C ) with |H| ≥ s, there is a normal H-interval

C [x1, x2] such that |C (x1, x2)| ≤ s − 1.

I Inserting vertices of H to C by removing paths—Find a cycle C ′

with V (C ′) ⊇ C [x2, x1] and 𝛼(H − V (C ′)) ≤ 𝛼(H)− 1.

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I How to use the Key Lemma?

By taking V0 = V (G ) in the Key Lemma, we see that if C is a

maximal cycle in G with |C | < k(s + 1), then for every induced

subgraph H of G − V (C ) with |H| ≥ s, there is a normal H-interval

C [x1, x2] such that |C (x1, x2)| ≤ s − 1.

I Inserting vertices of H to C by removing paths—Find a cycle C ′

with V (C ′) ⊇ C [x2, x1] and 𝛼(H − V (C ′)) ≤ 𝛼(H)− 1.

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Structures of graphs with independence number 𝛼

I Let G be a graph with independence number 𝛼.

� 𝛼 = 1 : G is Hamilton-connected.

� 𝛼 = 2 : either G has a hamiltonian cycle or V (G ) has a partition

(V1,V2) such that both G [V1] and G [V2] are cliques.

I equivalent form:

� 𝛼 = 1 : ∀ u = v ∈ V (G ), G has a (u, v)-path P such that

𝛼(G − V (P)) ≤ 𝛼(G )− 1.

(V1,V2) such that

𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).

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Structures of graphs with independence number 𝛼

I Let G be a graph with independence number 𝛼.

� 𝛼 = 1 : G is Hamilton-connected.

(V1,V2) such that both G [V1] and G [V2] are cliques.

I equivalent form:

� 𝛼 = 1 : ∀ u = v ∈ V (G ), G has a (u, v)-path P such that

𝛼(G − V (P)) ≤ 𝛼(G )− 1.

(V1,V2) such that

𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).

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Which structure of G − V (C ) is useful?

I Problem (asked by a referee of JGT)

c(G ) ≥ k(n + 𝛼− k)

𝛼− ck

for a constant ck?

I Remark: The fact that “G − V (C ) has a large subgraph H that is

hamiltonian” is not useful for the conclusion that

c(G ) ≥ k(n + 𝛼− k)

𝛼− ck

I In order to solve the above problem, we propose the following

conjecture.

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c(G ) ≥ k(n + 𝛼− k)

𝛼− ck

for a constant ck?

c(G ) ≥ k(n + 𝛼− k)

𝛼− ck

conjecture.

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c(G ) ≥ k(n + 𝛼− k)

𝛼− ck

for a constant ck?

c(G ) ≥ k(n + 𝛼− k)

𝛼− ck

conjecture.

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I Conjecture 17(Chen, Hu & Wu, 2009): For any graph G , one of the

following two statements holds.

(i) For any two distinct vertices u, v ∈ V (G ), there exists a

(u, v)-path P such that 𝛼(G − V (P)) ≤ 𝛼(G )− 1.

(ii) there is a non-trivial partition (V1,V2) of V (G ) such that

𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).

I Example: The sun graph S(Ct) doesn’t satisfies (i) and the cycle

C2m+1 doesn’t satisfies (ii).

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𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).

I Example: The sun graph S(Ct) doesn’t satisfies (i) and the cycle

C2m+1 doesn’t satisfies (ii).

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𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).

I Failed to find a 2-connected graph G that doesn’t satisfies (i), we

believe that (i) is true for every 2-connected graph. We prove the

following strong result by induction.

I Lemma 18 (Chen, Hu & Wu, 2010): Let G be a graph with

independence number 𝛼 and let u, v be two distinct vertices of G . If

𝜅(G ) ≥ 2, then G contains a (u, v)-path P such that

𝛼(G − V (P)) ≤ 𝛼(G − v)− 1.

Sketch of Proofs

𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).

𝛼(G − V (P)) ≤ 𝛼(G − v)− 1.

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𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).

𝛼(G − V (P)) ≤ 𝛼(G − v)− 1.

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independence number 𝛼 ≥ 2. If 𝜅(G ) ≤ 1, then there is a non-trivial

partition (V1,V2) of V (G ) such that 𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).

I By Lemmas 18 and 19, Conjecture 17 is true. Further, we have

I Theorem 20 Let G be a graph. Then, there exist two vertex disjoint

induced subgraph H1 and H2 of G such that

(i) V (G ) = V (H1) ∪ V (H2) and 𝛼(G ) = 𝛼(H1) + 𝛼(H2);

(ii) H2 = ∅ and for any two distinct vertices u, v ∈ V (H2), there

exists a (u, v)-path P in H2 such that 𝛼(H2 − V (P)) ≤ 𝛼(H2)− 1.

Sketch of Proofs

A low bound of c(G ) w.r.t a cycle C and an induced subgraph H of G − C

G [V0 − V (C )] with |H| > s − 1. If there is no normal H-interval

C [x1, x2] of C such that |C (x1, x2) ∩ V0| ≤ s − 1, then (i)

|V (C ) ∩ V0| ≥ ks; (ii) |V (C )| ≥ k(s + 1).

I By using the Key Lemma and Theorem 20, we can prove

I Theorem 21 (Chen, Hu & Wu, 2010). Let G be a k-connected

graph with k ≥ 2, let C be cycle in G and let H be a subgraph of

G − V (C ). Then, for every integer t ≥ 2, we have

c(G ) ≥ min{kt, |C |+ |H| − 𝛼(H)(t − 2)}.

Sketch of Proofs

|V (C ) ∩ V0| ≥ ks; (ii) |V (C )| ≥ k(s + 1).

c(G ) ≥ min{kt, |C |+ |H| − 𝛼(H)(t − 2)}.

Sketch of Proofs

|V (C ) ∩ V0| ≥ ks; (ii) |V (C )| ≥ k(s + 1).

c(G ) ≥ min{kt, |C |+ |H| − 𝛼(H)(t − 2)}.

Sketch of Proofs

Proof of Theorem 13

The following is a special form of Theorem 13.

I Theorem 22 (Chen, Hu & Wu, 2010) Let G be a k-connected

graph, k ≥ 2, of order n and V0 a nonempty subset of V (G ). Then

c(G ) ≥ min {|V0|, k ·max {f1(|V0|), ⌊f2(|V0|)⌋}}, wherefi (V0) =

|V0|+i(𝛼(G [V0])−k)𝛼(G [V0])

, i = 1, 2.

I Proof of Theorem 22:

� Find a cycle C in G such that V0 ⊆ V (C ) or

𝛼(G [V0]− V (C )) ≤ 𝛼(G [V0])− k (by using Kouider’s Theorem).

� If V0 ⊆ V (C ), then c(G ) ≥ |C | ≥ |V0|; If V0 ⊆ V (C ), then

𝛼(G [V0] > k. So, f1(V0) ≥ 2.

Sketch of Proofs

Proof of Theorem 13

The following is a special form of Theorem 13.

I Theorem 22 (Chen, Hu & Wu, 2010) Let G be a k-connected

graph, k ≥ 2, of order n and V0 a nonempty subset of V (G ). Then

c(G ) ≥ min {|V0|, k ·max {f1(|V0|), ⌊f2(|V0|)⌋}}, wherefi (V0) =

|V0|+i(𝛼(G [V0])−k)𝛼(G [V0])

, i = 1, 2.

I Proof of Theorem 22:

� Find a cycle C in G such that V0 ⊆ V (C ) or

𝛼(G [V0]− V (C )) ≤ 𝛼(G [V0])− k (by using Kouider’s Theorem).

� If V0 ⊆ V (C ), then c(G ) ≥ |C | ≥ |V0|; If V0 ⊆ V (C ), then

𝛼(G [V0] > k . So, f1(V0) ≥ 2.

Sketch of Proofs

I Proof of Theorem 22(continue):

� By using Theorem 21 with H = G [V0 − V (C )], we get

c(G ) ≥ min{kt, |C |+ |H| − 𝛼(H)(t − 2)}≥ min{kt, |V0| − (𝛼(G [V0])− k)(t − 2)}, (1)

where t is an integer with t ≥ 2.

� If c(G ) < kf1(V0), then by taking t = ⌈f1(V0)⌉ in (1), we have

kf1(V0) > |V0| − (𝛼(G [V0])− k)(⌈f1(V0)⌉ − 2)

≥ |V0| − (𝛼(G [V0])− k)(f1(V0)− 1),

which simplifies to f1(V0) >|V0|+𝛼(G [V0])−k

𝛼(G [V0]), a contradiction. Thus,

c(G ) ≥ kf1(V0). (2)

Sketch of Proofs

I Proof of Theorem 22(continue):

� If c(G ) < k⌊f2(V0)⌋, then by (2), ⌊f2(V0)⌋ > f1(V0) ≥ 2. By

taking t := ⌊f2(V0)⌋ in (1), we have

c(G ) ≥ |C |+ |H| − 𝛼(H)(⌊f2(V0)⌋ − 2)

≥ |V0| − (𝛼(G [V0])− k)(⌊f2(V0)⌋ − 2)

= (|V0|+ 2𝛼(G [V0])− 2k)− (𝛼(G [V0])− k)⌊f2(V0)⌋.

This together with c(G ) < k⌊f2(V0)⌋ implies that

⌊f2(V0)⌋ > |V0|+2𝛼(G [V0])−2k𝛼(G [V0])

, a contradiction.

Sketch of Proofs

Thank You Very Much For Your Attention!

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