Longest Cycles in k-connected Graphs with Given

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


Table of Contents

Definitions and Examples

Motivation and History of Fouquet-Jolivet Conjecture




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.







• 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.









• 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 .






• 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.







• 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.



Examples




Examples


n =



Examples


n = k + αm,



Examples


n = k + αm, α(G ) =



Examples


n = k + αm, α(G ) = α,



Examples


n = k + αm, α(G ) = α, κ(G ) =



Examples


n = k + αm, α(G ) = α, κ(G ) = k ,



Examples


n = k + αm, α(G ) = α, κ(G ) = k ,

c(G ) =



Examples


n = k + αm, α(G ) = α, κ(G ) = k ,

c(G ) = k(1 + m) = k(n+α−k)α

.



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.





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.





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.



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}.






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?



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 ).






c(G ) ≥k(n + α − k)

α


In 1982, Fournier proved the conjecture for α ∈ {k + 1, k + 2}.






c(G ) ≥k(n + α − k)

α



Two years later, he also proved it for k = 2.






c(G ) ≥k(n + α − k)

α



Two years later, he also proved it for k = 2.

In 2009, Manoussakis proved it for k = 3.



Sharpness of Fouquet-Jolivet Conjecture

Equality holds infinitley many often : G =Kk ∨ αKm forα ≥ k ≥ 2.





Sharpness examples of Fouquet-Jolivet Conjecture





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

.



WOW Theorem



WOW Theorem

Our main result is to show that Fouquet-Jolivet Conjecture is not aconjecture any more.



WOW Theorem

Our main result is to show that Fouquet-Jolivet Conjecture is not aconjecture any more.

WOW Theorem (2010+)


c(G ) ≥k(n + α − k)

α




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 .








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.








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.



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)}.



The Key Lemmas

Path Lemma


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.



The Key Lemmas

Path Lemma


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 .



The outline of the proof of Fouquet-Jolivet Conjecture

Path Lemma ⇒ Cycle Lemma ⇒

Multicycle Lemma ⇒ Fouquet-Jolivet Conjecture




Multicycle Lemma


i=0V (Ci )) ≤ α − k − l

(2) |V (Ci ) − ∪i−1j=0V (Cj)| ≤

|V (C0)|k

− 1 for 1 ≤ i ≤ l .




Multicycle Lemma


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.




Multicycle Lemma


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.



Fouquet-Jolivet Conjecture ⇒ Wow Theorem





Fouquet-Jolivet Conjecture ⇒ Wow Theorem


Multicycle Lemma ⇒ Wow Theorem


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Longest Cycles in k-connected Graphs with Given