Disproving the Normal Graph Conjecture
Ararat Harutyunyan (Inst. Math. Toulouse, U. Toulouse III)
Joint work with
Lucas Pastor (GSCOP Grenoble)
Stéphan Thomassé (LIP, ENS Lyon)
November 5, 2015
Introduction
Normal Graphs
Main Result: Disproving the conjecture
Perfect Graphs
Perfect Graphs
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∀G , χ(G ) ≥ ω(G ).
Perfect Graphs
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∀G , χ(G ) ≥ ω(G ).
G is perfect if for all induced H ⊂ G , χ(H) = ω(H).
Perfect Graphs
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∀G , χ(G ) ≥ ω(G ).
G is perfect if for all induced H ⊂ G , χ(H) = ω(H).
Examples: C5 is not perfect - χ(C5 ) = 3, ω(C5 ) = 2.
Perfect Graphs
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∀G , χ(G ) ≥ ω(G ).
G is perfect if for all induced H ⊂ G , χ(H) = ω(H).
Examples: C5 is not perfect - χ(C5 ) = 3, ω(C5 ) = 2.
Example: Kk is perfect.
The perfect graph theorems
The perfect graph theorems
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Theorem (WPGT, Lovász ’72)
G is perfect iff Ḡ is perfect.
The perfect graph theorems
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Theorem (WPGT, Lovász ’72)
G is perfect iff Ḡ is perfect.
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Theorem (SPGT, Chudnovsky, Robertson, Seymour, Thomas
2006)
G is perfect iff G does not contain an induced odd cycle of length
at least 5 or the complement of an induced odd cycle of length at
least 5.
Shannon capacity C (G )
Shannon capacity C (G )
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For a graph G : C (G ) = limn→∞
log ω(G n )
n
Shannon capacity C (G )
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For a graph G : C (G ) = limn→∞
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G n : nth co-normal power of G .
log ω(G n )
n
Shannon capacity C (G )
log ω(G n )
n
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For a graph G : C (G ) = limn→∞
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G n : nth co-normal power of G .
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Co-normal product G1 ∗ G2 : vertices V1 × V2 ,
(v1 , v2 ) ∼ (u1 , u2 ) if u1 ∼ v1 or v2 ∼ u2 .
Shannon capacity C (G )
log ω(G n )
n
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For a graph G : C (G ) = limn→∞
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G n : nth co-normal power of G .
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Co-normal product G1 ∗ G2 : vertices V1 × V2 ,
(v1 , v2 ) ∼ (u1 , u2 ) if u1 ∼ v1 or v2 ∼ u2 .
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For G perfect: ω(G n ) = ω(G )n
Normal Graphs
Normal Graphs
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Perfect graphs are not closed under co-normal products
(Körner, Longo ’73)
Normal Graphs
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Perfect graphs are not closed under co-normal products
(Körner, Longo ’73)
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G normal: ∃ two coverings of V , C, S,, s.t., every C ∈ C is a
clique, every S ∈ S is independent set and C ∩ S 6= ∅.
Normal Graphs
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Perfect graphs are not closed under co-normal products
(Körner, Longo ’73)
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G normal: ∃ two coverings of V , C, S,, s.t., every C ∈ C is a
clique, every S ∈ S is independent set and C ∩ S 6= ∅.
1,3,4
1,2,4
3
2
4
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1
More facts
More facts
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G is normal iff Ḡ is normal.
More facts
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G is normal iff Ḡ is normal.
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Perfect graphs are normal (Körner ’73).
More facts
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G is normal iff Ḡ is normal.
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Perfect graphs are normal (Körner ’73).
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Normal graphs are closed under co-normal products.
Cycles
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1
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2
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Cycles
1
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2
2
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C5 and C7 are NOT normal.
Cycles
1
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2
2
1
C5 and C7 are NOT normal.
Ck is normal, if k 6= 5, 7.
Cycles
1
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2
2
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C5 and C7 are NOT normal.
Ck is normal, if k 6= 5, 7.
No other minimal non-normal graphs known.
Normal Graph Conjecture
Normal Graph Conjecture
Normal Graph Conjecture
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Conjecture (DeSimone, Körner, ’99)
A graph with no C5 , C7 , C¯7 as induced subgraph is normal.
Disproving the normal graph conjecture
Theorem (Pastor, Thomassé, H. 2015+)
There exist graphs G of girth at least 8 which are not normal.
Disproving the normal graph conjecture
Theorem (Pastor, Thomassé, H. 2015+)
There exist graphs G of girth at least 8 which are not normal.
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The value 8 can be replaced by any number g .
Disproving the normal graph conjecture
Theorem (Pastor, Thomassé, H. 2015+)
There exist graphs G of girth at least 8 which are not normal.
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The value 8 can be replaced by any number g .
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The proof is probabilistic.
Random graphs
Random graphs
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Random graph Gn,p : n vertices, where any two are connected
with probability p, independently.
Random graphs
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Random graph Gn,p : n vertices, where any two are connected
with probability p, independently.
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We take p = n−0.9 .
Random graphs
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Random graph Gn,p : n vertices, where any two are connected
with probability p, independently.
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We take p = n−0.9 .
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Caution: can have cycles of lenght up to 7!
Random graphs
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Random graph Gn,p : n vertices, where any two are connected
with probability p, independently.
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We take p = n−0.9 .
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Caution: can have cycles of lenght up to 7!
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Xk : number of cycles of length k.
k
k k
k/10 < n0.7 .
E[Xk ] ≤ kn (k−1)!
2 p <n p =n
Random graphs
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Random graph Gn,p : n vertices, where any two are connected
with probability p, independently.
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We take p = n−0.9 .
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Caution: can have cycles of lenght up to 7!
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Xk : number of cycles of length k.
k
k k
k/10 < n0.7 .
E[Xk ] ≤ kn (k−1)!
2 p <n p =n
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Markov’s Inequality: Pr [Xk > 2n0.7 ] < 1/2
Random graphs
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Random graph Gn,p : n vertices, where any two are connected
with probability p, independently.
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We take p = n−0.9 .
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Caution: can have cycles of lenght up to 7!
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Xk : number of cycles of length k.
k
k k
k/10 < n0.7 .
E[Xk ] ≤ kn (k−1)!
2 p <n p =n
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Markov’s Inequality: Pr [Xk > 2n0.7 ] < 1/2
Remove one vertex from each of 2n0.7 cycles to destroy all
cycles length < 8.
Other facts
Other facts
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Pr [α(Gn,p ) ≥ x] ≤
n
x
x
(1 − p)(2) .
Other facts
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n
(x2) .
x (1 − p)
O(n0.9 log n).
Pr [α(Gn,p ) ≥ x] ≤
Whp α(Gn,p ) =
Other facts
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n
(x2) .
x (1 − p)
O(n0.9 log n).
Pr [α(Gn,p ) ≥ x] ≤
Whp α(Gn,p ) =
Degrees: Whp almost every vertex has degree Ω(n0.1 )
Overview of the proof
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G girth 8 and normal, then we may assume the clique cover
induces vertex-disjoint stars.
u
v
Overview of the proof
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G girth 8 and normal, then we may assume the clique cover
induces vertex-disjoint stars.
u
v
Obtaining a large independent set
Obtaining a large independent set
u∈S u
1
∈
/S
u2
u3
∈
/S
u4
∈
/S
Thank You