Longest Cycles in k-connected Graphs with Given ...
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Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Longest Cycles in k-connected Graphs with Given
Independence Number
Suil O, Douglas B. West, and Hehui Wu
University of Illinois at Urbana-Champaign
AMS Meeting (Joint Mathematics Meetings) 2011
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Table of Contents
Definitions and Examples
Motivation and History of Fouquet-Jolivet Conjecture
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
• d(v): the degree of vertex v .
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
• d(v): the degree of vertex v .
• δ(G ): the minimum degree of a graph G .
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
• d(v): the degree of vertex v .
• δ(G ): the minimum degree of a graph G .
• The join of G and H, denoted G ∨H, is the graph obtained fromG and H by joining every vertex of G to every vertex of H.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
• d(v): the degree of vertex v .
• δ(G ): the minimum degree of a graph G .
• The join of G and H, denoted G ∨H, is the graph obtained fromG and H by joining every vertex of G to every vertex of H.
• A graph G is said to be k-connected when there does not exist aset of (k − 1)-vertices whose removal disconnects the graph.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
• κ(G ): the connectivity of a graph G , the largest positive integerk such that G is k-connected.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
• κ(G ): the connectivity of a graph G , the largest positive integerk such that G is k-connected.
• c(G ): the circumference of a graph G , the length of a largestcycle in G .
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
• κ(G ): the connectivity of a graph G , the largest positive integerk such that G is k-connected.
• c(G ): the circumference of a graph G , the length of a largestcycle in G . A graph G is said to be hamiltonian when c(G ) = n.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
• κ(G ): the connectivity of a graph G , the largest positive integerk such that G is k-connected.
• c(G ): the circumference of a graph G , the length of a largestcycle in G . A graph G is said to be hamiltonian when c(G ) = n.
• A vertex subset S ⊆ V (G ) is said to be independent such thatno two vertices are adjacent.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Definitions and Examples
Kk ∨ αKm for α ≥ k ≥ 2
• κ(G ): the connectivity of a graph G , the largest positive integerk such that G is k-connected.
• c(G ): the circumference of a graph G , the length of a largestcycle in G . A graph G is said to be hamiltonian when c(G ) = n.
• A vertex subset S ⊆ V (G ) is said to be independent such thatno two vertices are adjacent.
• α(G ): the independence number of a graph G , the maximumsize of an independent set of vertices.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Examples
Kk ∨ αKm for α ≥ k ≥ 2
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Examples
Kk ∨ αKm for α ≥ k ≥ 2
n =
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Examples
Kk ∨ αKm for α ≥ k ≥ 2
n = k + αm,
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Examples
Kk ∨ αKm for α ≥ k ≥ 2
n = k + αm, α(G ) =
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Examples
Kk ∨ αKm for α ≥ k ≥ 2
n = k + αm, α(G ) = α,
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Examples
Kk ∨ αKm for α ≥ k ≥ 2
n = k + αm, α(G ) = α, κ(G ) =
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Examples
Kk ∨ αKm for α ≥ k ≥ 2
n = k + αm, α(G ) = α, κ(G ) = k ,
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Examples
Kk ∨ αKm for α ≥ k ≥ 2
n = k + αm, α(G ) = α, κ(G ) = k ,
c(G ) =
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Examples
Kk ∨ αKm for α ≥ k ≥ 2
n = k + αm, α(G ) = α, κ(G ) = k ,
c(G ) = k(1 + m) = k(n+α−k)α
.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Sufficient Conditions for Hamiltonicity
In 1952, Dirac proved that if G is a simple graph with n vertices(n ≥ 3) and δ(G ) ≥ n
2 , then c(G ) = n.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Sufficient Conditions for Hamiltonicity
In 1952, Dirac proved that if G is a simple graph with n vertices(n ≥ 3) and δ(G ) ≥ n
2 , then c(G ) = n.
More generally, in 1960, Ore proved that if a graph G satisfies theproperty that d(u) + d(v) ≥ n whenever uv /∈ E (G ), thenc(G ) = n.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Sufficient Conditions for Hamiltonicity
In 1952, Dirac proved that if G is a simple graph with n vertices(n ≥ 3) and δ(G ) ≥ n
2 , then c(G ) = n.
More generally, in 1960, Ore proved that if a graph G satisfies theproperty that d(u) + d(v) ≥ n whenever uv /∈ E (G ), thenc(G ) = n.
In 1972, Chvatal and Erdos showed that if κ(G ) ≥ α(G ) for agraph G , then c(G ) = n.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Longest Cycle Versions for 2-connected Graphs
In 1952, Dirac also proved that if G is a 2-connected graph with n
vertices, then c(G ) ≥ min{n, 2δ(G )}.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Longest Cycle Versions for 2-connected Graphs
In 1952, Dirac also proved that if G is a 2-connected graph with n
vertices, then c(G ) ≥ min{n, 2δ(G )}.
More generally, Bondy(1971), and Bermond and Linial(1976)proved that if a 2-connected graph G satisfies the property thatd(u) + d(v) ≥ s whenever uv /∈ E (G ), then c(G ) ≥ min{n, s}.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Longest Cycle Versions for 2-connected Graphs
In 1952, Dirac also proved that if G is a 2-connected graph with n
vertices, then c(G ) ≥ min{n, 2δ(G )}.
More generally, Bondy(1971), and Bermond and Linial(1976)proved that if a 2-connected graph G satisfies the property thatd(u) + d(v) ≥ s whenever uv /∈ E (G ), then c(G ) ≥ min{n, s}.
Now, is there a longest cycle version of Chvatal and Erdostheorem?
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Fouquet-Jolivet Conjecture
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Fouquet-Jolivet Conjecture
Conjecture (Fouquet-Jolivet 1976)
If κ(G ) ≤ α(G ), then
c(G ) ≥k(n + α − k)
α
, where k = κ(G ), and α = α(G ).
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Fouquet-Jolivet Conjecture
Conjecture (Fouquet-Jolivet 1976)
If κ(G ) ≤ α(G ), then
c(G ) ≥k(n + α − k)
α
, where k = κ(G ), and α = α(G ).
In 1982, Fournier proved the conjecture for α ∈ {k + 1, k + 2}.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Fouquet-Jolivet Conjecture
Conjecture (Fouquet-Jolivet 1976)
If κ(G ) ≤ α(G ), then
c(G ) ≥k(n + α − k)
α
, where k = κ(G ), and α = α(G ).
In 1982, Fournier proved the conjecture for α ∈ {k + 1, k + 2}.
Two years later, he also proved it for k = 2.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Fouquet-Jolivet Conjecture
Conjecture (Fouquet-Jolivet 1976)
If κ(G ) ≤ α(G ), then
c(G ) ≥k(n + α − k)
α
, where k = κ(G ), and α = α(G ).
In 1982, Fournier proved the conjecture for α ∈ {k + 1, k + 2}.
Two years later, he also proved it for k = 2.
In 2009, Manoussakis proved it for k = 3.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Sharpness of Fouquet-Jolivet Conjecture
Equality holds infinitley many often : G =Kk ∨ αKm forα ≥ k ≥ 2.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Sharpness of Fouquet-Jolivet Conjecture
Equality holds infinitley many often : G =Kk ∨ αKm forα ≥ k ≥ 2.
Sharpness examples of Fouquet-Jolivet Conjecture
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Sharpness of Fouquet-Jolivet Conjecture
Equality holds infinitley many often : G =Kk ∨ αKm forα ≥ k ≥ 2.
Sharpness examples of Fouquet-Jolivet Conjecture
n=k + αm, α(G ) =α, κ(G )=k , c(G ) =k(1 + m)= k(n+α−k)α
.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
WOW Theorem
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
WOW Theorem
Our main result is to show that Fouquet-Jolivet Conjecture is not aconjecture any more.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
WOW Theorem
Our main result is to show that Fouquet-Jolivet Conjecture is not aconjecture any more.
WOW Theorem (2010+)
If κ(G ) ≤ α(G ), then
c(G ) ≥k(n + α − k)
α
, where k = κ(G ), and α = α(G ).
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
History of Fouquet-Jolivet Conjecture
When Fournier and Manousakis proved for the cases k = 2 andk = 3, respectively, both of them used the special cases (k = 2and k = 3) of the following conjecture.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
History of Fouquet-Jolivet Conjecture
When Fournier and Manousakis proved for the cases k = 2 andk = 3, respectively, both of them used the special cases (k = 2and k = 3) of the following conjecture.
Conjecture(Chen-Chen-Liu)
If C1 and C2 are distinct cycles in a k-connected graph G , thenthere are distinct cycles C ′
1 and C ′2 in G such that V (C1)∪V (C2) ⊆
V (C ′1) ∪ V (C ′
2) and |V (C ′1) ∩ V (C ′
2)| ≥ k .
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
History of Fouquet-Jolivet Conjecture
When Fournier and Manousakis proved for the cases k = 2 andk = 3, respectively, both of them used the special cases (k = 2and k = 3) of the following conjecture.
Conjecture(Chen-Chen-Liu)
If C1 and C2 are distinct cycles in a k-connected graph G , thenthere are distinct cycles C ′
1 and C ′2 in G such that V (C1)∪V (C2) ⊆
V (C ′1) ∪ V (C ′
2) and |V (C ′1) ∩ V (C ′
2)| ≥ k .
Chen, Hu, and Wu(2010+) proved that Chen-Chen-Liu conjectureimplies Fouquet-Jolivet conjecture.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
History of Fouquet-Jolivet Conjecture
When Fournier and Manousakis proved for the cases k = 2 andk = 3, respectively, both of them used the special cases (k = 2and k = 3) of the following conjecture.
Conjecture(Chen-Chen-Liu)
If C1 and C2 are distinct cycles in a k-connected graph G , thenthere are distinct cycles C ′
1 and C ′2 in G such that V (C1)∪V (C2) ⊆
V (C ′1) ∪ V (C ′
2) and |V (C ′1) ∩ V (C ′
2)| ≥ k .
Chen, Hu, and Wu(2010+) proved that Chen-Chen-Liu conjectureimplies Fouquet-Jolivet conjecture.
Without using Chen-Chen-Liu Conjecture, we proveFouquet-Jolivet Conjecture.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
The Key Lemmas
Path Lemma
If H is a subgraph of a k-connected graph G , and u, v ∈ V (G ), thenG has a u, v -path P with α(H −V (P)) ≤ max{0, α(H)− (k − 1)}.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
The Key Lemmas
Path Lemma
If H is a subgraph of a k-connected graph G , and u, v ∈ V (G ), thenG has a u, v -path P with α(H −V (P)) ≤ max{0, α(H)− (k − 1)}.
Cycle Lemma
If H and C are disjoint subgraphs of a k-connected graph G , withC being a cycle of length ≥ k , then G has a cycle C ′ such that|V (C ) − V (C ′)| ≤ |V (C)|
k− 1 and α(H − V (C ′)) ≤ α(H) − 1.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
The Key Lemmas
Path Lemma
If H is a subgraph of a k-connected graph G , and u, v ∈ V (G ), thenG has a u, v -path P with α(H −V (P)) ≤ max{0, α(H)− (k − 1)}.
Cycle Lemma
If H and C are disjoint subgraphs of a k-connected graph G , withC being a cycle of length ≥ k , then G has a cycle C ′ such that|V (C ) − V (C ′)| ≤ |V (C)|
k− 1 and α(H − V (C ′)) ≤ α(H) − 1.
Multicycle Lemma
If G is a k-connected graph with α(G ) = α, and 0 ≤ l ≤ α − k ,then there exists cycles C0, · · · , Cl satisfying the following :(1) α(G − ∪l
i=0V (Ci )) ≤ α − k − l
(2) |V (Ci ) − ∪i−1j=0V (Cj)| ≤
|V (C0)|k
− 1 for 1 ≤ i ≤ l .
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
The outline of the proof of Fouquet-Jolivet Conjecture
Path Lemma ⇒ Cycle Lemma ⇒
Multicycle Lemma ⇒ Fouquet-Jolivet Conjecture
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Multicycle Lemma ⇒ Fouquet-Jolivet Conjecture
Multicycle Lemma
If G is a k-connected graph with α(G ) = α, and 0 ≤ l ≤ α − k ,then there exists cycles C0, · · · , Cl satisfying the following :(1) α(G − ∪l
i=0V (Ci )) ≤ α − k − l
(2) |V (Ci ) − ∪i−1j=0V (Cj)| ≤
|V (C0)|k
− 1 for 1 ≤ i ≤ l .
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Multicycle Lemma ⇒ Fouquet-Jolivet Conjecture
Multicycle Lemma
If G is a k-connected graph with α(G ) = α, and 0 ≤ l ≤ α − k ,then there exists cycles C0, · · · , Cl satisfying the following :(1) α(G − ∪l
i=0V (Ci )) ≤ α − k − l
(2) |V (Ci ) − ∪i−1j=0V (Cj)| ≤
|V (C0)|k
− 1 for 1 ≤ i ≤ l .
WOW Theorem(2010+)
Fouquet-Jolivet Conjecture is true.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Multicycle Lemma ⇒ Fouquet-Jolivet Conjecture
Multicycle Lemma
If G is a k-connected graph with α(G ) = α, and 0 ≤ l ≤ α − k ,then there exists cycles C0, · · · , Cl satisfying the following :(1) α(G − ∪l
i=0V (Ci )) ≤ α − k − l
(2) |V (Ci ) − ∪i−1j=0V (Cj)| ≤
|V (C0)|k
− 1 for 1 ≤ i ≤ l .
WOW Theorem(2010+)
Fouquet-Jolivet Conjecture is true.
Proof. Consider l = α − k in Multicle Lemma.Thus C0, · · · , Cl cover V (G ) by (1).By (2), n = |V (C0)| +
∑li=1 |V (Ci ) − ∪i−1
j=0V (Cj)| ≤
|V (C0)| + (α − k)(
|V (C0)|k
− 1)
.
The inequality simplies to |V (C0)| ≥k(n+α−k)
α.
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence
Definitions and ExamplesMotivation and History of Fouquet-Jolivet Conjecture
Fouquet-Jolivet Conjecture ⇒ Wow Theorem
Path Lemma ⇒ Cycle Lemma ⇒
Multicycle Lemma ⇒ Fouquet-Jolivet Conjecture
O, West, and Wu Longest Cycles in k-connected Graphs with Given Independence