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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
Notations and DefinitionsMain Results
Sketch of Proofs
Dedicated to Professor Tian’s 70th birthday
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
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
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
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
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
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
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
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
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
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
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
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
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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
then G is hamiltonian.
I Theorem D (Fan,1984) Let G be a k-connected graph. If
𝜇(G ) ≥ n/2, then G is hamiltonian.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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
then G is hamiltonian.
I Theorem D (Fan,1984) Let G be a k-connected graph. If
𝜇(G ) ≥ n/2, then G is hamiltonian.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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
then G is hamiltonian.
I Theorem D (Fan,1984) Let G be a k-connected graph. If
𝜇(G ) ≥ n/2, then G is hamiltonian.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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
then G is hamiltonian.
I Theorem D (Fan,1984) Let G be a k-connected graph. If
𝜇(G ) ≥ n/2, then G is hamiltonian.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
Long Cycles Involving Independence Numbers
I Theorem 1 (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
then G is hamiltonian.
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)
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
Long Cycles Involving Independence Numbers
I Theorem 1 (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
then G is hamiltonian.
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)
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
Long Cycles Involving Independence Numbers
I Theorem 1 (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
then G is hamiltonian.
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)
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)
2 .
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)
2 .
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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
17).
I We proved Conjecture 17 on March 2010 and get the following two
results:
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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
17).
I We proved Conjecture 17 on March 2010 and get the following two
results:
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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
17).
I We proved Conjecture 17 on March 2010 and get the following two
results:
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)},
I Theorem 7 (Chen, Hu & Wu, March 2010) Let G be a k-connected
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)
p,
where p = ⌊𝛼 ∖ k⌋ and k0 is an integer such that Conjecture 8 is
true for k = k0.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)},
I Theorem 7 (Chen, Hu & Wu, March 2010) Let G be a k-connected
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)
p,
where p = ⌊𝛼 ∖ k⌋ and k0 is an integer such that Conjecture 8 is
true for k = k0.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)
𝛼.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)
𝛼.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)
𝛼.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)
𝛼.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I By use Kouider’s Theorem one times and Theorem 6, We proved
the following generalization of Theorem 10.
I Theorem 11 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I By use Kouider’s Theorem one times and Theorem 6, We proved
the following generalization of Theorem 10.
I Theorem 11 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I By use Kouider’s Theorem one times and Theorem 6, We proved
the following generalization of Theorem 10.
I Theorem 11 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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 .
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)
𝛼.
I Theorem 11 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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 12 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
of order n and independence number 𝛼. If 𝛼 ≥ k ≥ 2, then
c(G ) ≥ min
{n,max
{k(n + 𝛼− k)
𝛼, k
⌊n + 2𝛼− 2k
𝛼
⌋}}.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)
𝛼.
I Theorem 11 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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 12 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
of order n and independence number 𝛼. If 𝛼 ≥ k ≥ 2, then
c(G ) ≥ min
{n,max
{k(n + 𝛼− k)
𝛼, k
⌊n + 2𝛼− 2k
𝛼
⌋}}.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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)
𝛼.
I Theorem 11 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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 12 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
of order n and independence number 𝛼. If 𝛼 ≥ k ≥ 2, then
c(G ) ≥ min
{n,max
{k(n + 𝛼− k)
𝛼, k
⌊n + 2𝛼− 2k
𝛼
⌋}}.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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
𝛼
⌋> k(n+𝛼−k)
𝛼
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I Theorem 12 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
of order n and independence number 𝛼. If 𝛼 ≥ k ≥ 2, then
c(G ) ≥ min
{n,max
{k(n + 𝛼− k)
𝛼, k
⌊n + 2𝛼− 2k
𝛼
⌋}}.
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).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I Theorem 12 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
of order n and independence number 𝛼. If 𝛼 ≥ k ≥ 2, then
c(G ) ≥ min
{n,max
{k(n + 𝛼− k)
𝛼, k
⌊n + 2𝛼− 2k
𝛼
⌋}}.
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).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I Theorem 12 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
of order n and independence number 𝛼. If 𝛼 ≥ k ≥ 2, then
c(G ) ≥ min
{n,max
{k(n + 𝛼− k)
𝛼, k
⌊n + 2𝛼− 2k
𝛼
⌋}}.
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).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I The following is a common generalization of Theorems 10-12.
I Theorem 13 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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 ).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I The following is a common generalization of Theorems 10-12.
I Theorem 13 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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 ).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I The following is a common generalization of Theorems 10-12.
I Theorem 13 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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 ).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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 .
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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 .
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I We get the following theorem.
I Theorem 16 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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)
⌋}.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
I We get the following theorem.
I Theorem 16 (Chen, Hu & Wu, 2010) Let G be a k-connected graph
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)
⌋}.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
Classic results for hamiltonian graphsLong Cycles Involving Independence NumbersLong Cycles Intersecting a given subgraph
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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}.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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
C .
I Question: What happens if m = 0 and H is an induced subgraph of
G − V (C )?
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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
C .
I Question: What happens if m = 0 and H is an induced subgraph of
G − V (C )?
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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
C .
I Question: What happens if m = 0 and H is an induced subgraph of
G − V (C )?
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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.
� 𝛼 = 2 : either G has a hamiltonian cycle or V (G ) has a partition
(V1,V2) such that
𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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.
� 𝛼 = 2 : either G has a hamiltonian cycle or V (G ) has a partition
(V1,V2) such that
𝛼(G ) = 𝛼(G [V1]) + 𝛼(G [V2]).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
Which structure of G − V (C ) is useful?
I Problem (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 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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
Which structure of G − V (C ) is useful?
I Problem (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 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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
Which structure of G − V (C ) is useful?
I Problem (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 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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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).
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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 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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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 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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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 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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
I Lemma 19 (Chen, Hu & Wu, 2009): Let G be a graph with
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
I Lemma 19 (Chen, Hu & Wu, 2009): Let G be a graph with
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
I Lemma 19 (Chen, Hu & Wu, 2009): Let G be a graph with
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
A low bound of c(G ) w.r.t a cycle C and an induced subgraph H of G − C
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 − 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)}.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
A low bound of c(G ) w.r.t a cycle C and an induced subgraph H of G − C
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 − 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)}.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
A low bound of c(G ) w.r.t a cycle C and an induced subgraph H of G − C
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 − 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)}.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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)
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
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.
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture
Notations and DefinitionsMain Results
Sketch of Proofs
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 .
Thank You Very Much For Your Attention!
Zhiquan Hu Extension of Fouquet-Jolivet’s Conjecture