The Lovász Conjecture and extensions
Graph colourings, spaces of edges and spaces of circuits
Carsten Schultz
Technische Universität Berlin
Workshop on Topological Methods in Combinatorics
Stockholm 2006
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
1 / 20
Outline
1
Main result
2
Examples and Consequences
Cycles in complete and cyclic graphs
Graph colouring obstructions
3
Bits of the proof
4
Preview of further results
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
2 / 20
Outline
1
Main result
2
Examples and Consequences
Cycles in complete and cyclic graphs
Graph colouring obstructions
3
Bits of the proof
4
Preview of further results
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
3 / 20
Hom-posets and -complexes
A graph homomorphism G → H is a function V (G ) → V (H) that
preserves the adjacency relation.
A multi-homomorphism φ : G → H is
φ : V (G ) → P(V (H))\ {Ø}
such that every choice function for φ is a homomorphism
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
4 / 20
Hom-posets and -complexes
A graph homomorphism G → H is a function V (G ) → V (H) that
preserves the adjacency relation.
A multi-homomorphism φ : G → H is
φ : V (G ) → P(V (H))\ {Ø}
such that every choice function for φ is a homomorphism
Hom(G , H) is the poset of all multi-homomorphisms from G to H.
The composition map
∗ : Hom(G , G 0 ) × Hom(G 0 , G 00 ) −−→ Hom(G , G 00 )
(φ ∗ ρ)(v ) := ρ[φ(v )]
is associative and order-preserving.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
4 / 20
Hom-posets and -complexes
A graph homomorphism G → H is a function V (G ) → V (H) that
preserves the adjacency relation.
A multi-homomorphism φ : G → H is
φ : V (G ) → P(V (H))\ {Ø}
such that every choice function for φ is a homomorphism
Hom(G , H) is the poset of all multi-homomorphisms from G to H.
The composition map
∗ : Hom(G , G 0 ) × Hom(G 0 , G 00 ) −−→ Hom(G , G 00 )
(φ ∗ ρ)(v ) := ρ[φ(v )]
is associative and order-preserving.
α : G → G with α2 = id makes |Hom(G , H)| into a free Z2 -space,
if α flips an edge and H is loop-free.
In particular, |Hom(K2 , H)| is a free Z2 -space.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
4 / 20
Hom-posets and -complexes
A graph homomorphism G → H is a function V (G ) → V (H) that
preserves the adjacency relation.
A multi-homomorphism φ : G → H is
φ : V (G ) → P(V (H))\ {Ø}
such that every choice function for φ is a homomorphism
Hom(G , H) is the poset of all multi-homomorphisms from G to H.
The composition map
∗ : Hom(G , G 0 ) × Hom(G 0 , G 00 ) −−→ Hom(G , G 00 )
(φ ∗ ρ)(v ) := ρ[φ(v )]
is associative and order-preserving.
α : G → G with α2 = id makes |Hom(G , H)| into a free Z2 -space,
if α flips an edge and H is loop-free.
In particular, |Hom(K2 , H)| is a free Z2 -space.
|Hom(K2 , Kn )| ≈Z2 Sn−2 .
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
4 / 20
Lovász’ Theorem and Conjecture
Theorem (Lovász ’78)
Let G be a graph. Then
χ(G ) ≥ indZ2 |Hom(K2 , G )| + 2.
Theorem (Babson & Kozlov ’04)
Let G be a graph, r ≥ 1. Then
χ(G ) ≥ coindZ2 |Hom(C2r +1 , G )| + 3.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
5 / 20
Lovász’ Theorem and Conjecture
Theorem (Lovász ’78)
Let G be a graph. Then
χ(G ) ≥ indZ2 |Hom(K2 , G )| + 2.
Theorem (Babson & Kozlov ’04)
Let G be a graph, r ≥ 1. Then
χ(G ) ≥ coindZ2 |Hom(C2r +1 , G )| + 3.
Question
What is the relationship between Hom(K2 , G ) and Hom(C2r +1 , G )?
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
5 / 20
Spaces of cycles of arbitrary lengths
The main theorem
There is a homomorphism Cm+2 →Z2 Cm .
This induces Hom(Cm , G ) →Z2 Hom(Cm+2 , G ).
We can consider colimr |Hom(C2r +1 , G )|.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
6 / 20
Spaces of cycles of arbitrary lengths
The main theorem
There is a homomorphism Cm+2 →Z2 Cm .
This induces Hom(Cm , G ) →Z2 Hom(Cm+2 , G ).
We can consider colimr |Hom(C2r +1 , G )|.
Theorem
colim|Hom(C2r +1 , G )| 'Z2 MapZ2 (S1b , |Hom(K2 , G )|)
r
colim|Hom(C2r , G )| 'Z2 Map(S1b , |Hom(K2 , G )|)
r
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
6 / 20
Outline
1
Main result
2
Examples and Consequences
Cycles in complete and cyclic graphs
Graph colouring obstructions
3
Bits of the proof
4
Preview of further results
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
7 / 20
Spaces of cycles of arbitrary lengths
in complete graphs
Since |Hom(K2 , Kn )| ≈Z2 Sn−2 :
colimr |Hom(C2r , Kn )| ' Map(S1 , Sn−2 ) free loop space of a sphere.
colimr |Hom(C2r +1 , Kn )| ' MapZ2 (S1 , Sn−2 ).
Kozlov, D. N.
Cohomology of colorings of cycles, 2005.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
math.AT/0507117.
Stockholm ’06
8 / 20
Spaces of cycles of arbitrary lengths
in complete graphs
Since |Hom(K2 , Kn )| ≈Z2 Sn−2 :
colimr |Hom(C2r , Kn )| ' Map(S1 , Sn−2 ) free loop space of a sphere.
colimr |Hom(C2r +1 , Kn )| ' MapZ2 (S1 , Sn−2 ).
There is a canonical embedding map
V2,n−1 := (x, y ) ∈ Sn−2 : hx, y i = 0 −−→ MapZ2 (S1 , Sn−2 )
“Start at x, follow great circle through y .”
Kozlov, D. N.
Cohomology of colorings of cycles, 2005.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
math.AT/0507117.
Stockholm ’06
8 / 20
Spaces of cycles of arbitrary lengths
in complete graphs
Since |Hom(K2 , Kn )| ≈Z2 Sn−2 :
colimr |Hom(C2r , Kn )| ' Map(S1 , Sn−2 ) free loop space of a sphere.
colimr |Hom(C2r +1 , Kn )| ' MapZ2 (S1 , Sn−2 ).
There is a canonical embedding map
V2,n−1 := (x, y ) ∈ Sn−2 : hx, y i = 0 −−→ MapZ2 (S1 , Sn−2 )
“Start at x, follow great circle through y .”
Theorem (S, conjectured by Csorba)
|Hom(C5 , Kn )| ≈ V2,n−1 .
Kozlov, D. N. Cohomology of colorings of cycles, 2005.
math.AT/0507117.
Csorba, P. Non-tidy Spaces and Graph Colorings.
Ph.D. thesis, ETH Zürich, 2005.
Csorba, P. and Lutz, F. H. Graph coloring manifolds, 2005.
math.CO/0510177.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
8 / 20
Spaces of cycles of arbitrary lengths
in cyclic graphs
Proposition
|Hom(K2 , C2r +1 )| ≈Z2
|Hom(K2 , C2r )| ≈Z2
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
9 / 20
Spaces of cycles of arbitrary lengths
in cyclic graphs
Proposition
|Hom(K2 , C2r +1 )| ≈Z2
|Hom(K2 , C2r )| ≈Z2
Corollary
colim|Hom(C2r , Cm )| ' Map(S1 , |Hom(K2 , Cm )|) '
a
r
S1
Z
colim|Hom(C2r +1 , Cm )| ' MapZ2 (S1 , |Hom(K2 , Cm )|) '
r
(`
1
ZS ,
Ø,
m odd,
m even.
Čukić, S. L. and Kozlov, D. N. The homotopy type of complexes of graph
homomorphisms between cycles.
Discrete Comp. Geometry .
In press, math.CO/0408015.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
9 / 20
Free Z2 -spaces
Definition
Let X be a free Z2 -space. We define
n
o
indZ2 X := min k : There is a Z2 -map X → Sk ,
n
o
coindZ2 X := max k : There is a Z2 -map Sk → X ,
n
o
cohom-indZ2 X := max k : f ∗ (γ k ) 6= 0 ,
where f : X /Z2 → RP∞ is classifying and H ∗ (RP∞ ; Z2 ) = Z2 [γ].
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
10 / 20
Free Z2 -spaces
Definition
Let X be a free Z2 -space. We define
n
o
indZ2 X := min k : There is a Z2 -map X → Sk ,
n
o
coindZ2 X := max k : There is a Z2 -map Sk → X ,
n
o
cohom-indZ2 X := max k : f ∗ (γ k ) 6= 0 ,
where f : X /Z2 → RP∞ is classifying and H ∗ (RP∞ ; Z2 ) = Z2 [γ].
Properties
conn X + 1 ≤ coindZ2 X ≤ cohom-indZ2 X ≤ indZ2 X
coindZ2 Sn = indZ2 Sn = n
If there is X →Z2 Y then x-indZ2 X ≤ x-indZ2 Y .
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
10 / 20
Some topological facts
Let X , Y be free Z2 -spaces and S1b the 1-sphere with the Z2 -operations
τ · (x0 , x1 ) := (−x0 , −x1 ),
(x0 , xk ) · τ := (−x0 , x1 ).
There is an adjunction
TopZ2 (Y , MapZ2 (S1b , X )) ∼
= TopZ2 (S1b ×Z2 Y , X ).
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
11 / 20
Some topological facts
Let X , Y be free Z2 -spaces and S1b the 1-sphere with the Z2 -operations
τ · (x0 , x1 ) := (−x0 , −x1 ),
(x0 , xk ) · τ := (−x0 , x1 ).
There is an adjunction
TopZ2 (Y , MapZ2 (S1b , X )) ∼
= TopZ2 (S1b ×Z2 Y , X ).
We obtain inequalities
cohom-indZ2 X + 1 ≤ cohom-indZ2 (S1b ×Z2 X ),
indZ2 (S1b ×Z2 X ) ≤ indZ2 X + 1,
cohom-indZ2 MapZ2 (S1b , X ) + 1 ≤ cohom-indZ2 X ,
coindZ2 X ≤ coindZ2 MapZ2 (S1b , X ) + 1.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
11 / 20
Spaces of cycles of arbitrary lengths
Consequences
Theorems (Lovász ’78, Babson & Kozlov ’04)
χ(G ) ≥ indZ2 |Hom(K2 , G )| + 2.
χ(G ) ≥ coindZ2 |Hom(C2r +1 , G )| + 3.
Theorem
colimr |Hom(C2r +1 , G )| 'Z2 MapZ2 (S1b , |Hom(K2 , G )|).
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
12 / 20
Spaces of cycles of arbitrary lengths
Consequences
Theorems (Lovász ’78, Babson & Kozlov ’04)
χ(G ) ≥ indZ2 |Hom(K2 , G )| + 2.
χ(G ) ≥ coindZ2 |Hom(C2r +1 , G )| + 3.
Theorem
colimr |Hom(C2r +1 , G )| 'Z2 MapZ2 (S1b , |Hom(K2 , G )|).
Corollary
cohom-indZ2 |Hom(C2r +1 , G )| + 1 ≤ cohom-indZ2 |Hom(K2 , G )|.
lim coindZ2 |Hom(C2r +1 , G )| + 1 ≥ coindZ2 |Hom(K2 , G )|.
r →∞
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
12 / 20
Outline
1
Main result
2
Examples and Consequences
Cycles in complete and cyclic graphs
Graph colouring obstructions
3
Bits of the proof
4
Preview of further results
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
13 / 20
Hom(K2 , C2r +1 )
A closer look.
Reminder
We want to compare Hom(K2 , G ) and Hom(C2r +1 , G ).
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
14 / 20
Hom(K2 , C2r +1 )
A closer look.
C5
|Hom(K2 , C5 )|
Definition
S1b is the 1-sphere with the
Z2 -operations
τ · (x0 , x1 ) := (−x0 , −x1 ),
(x0 , xk ) · τ := (−x0 , x1 ).
Proposition
|Hom(K2 , C2r +1 )| ≈Z2 ×Z2 S1b
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
14 / 20
The easy part
The composition map
Hom(K2 , C2r +1 ) × Hom(C2r +1 , G ) −−→ Hom(K2 , G )
yields
S1b ×Z2 |Hom(C2r +1 , G )| −−→Z2 |Hom(K2 , G )|.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
15 / 20
The easy part
The composition map
Hom(K2 , C2r +1 ) × Hom(C2r +1 , G ) −−→ Hom(K2 , G )
yields
S1b ×Z2 |Hom(C2r +1 , G )| −−→Z2 |Hom(K2 , G )|.
Direct consequence:
cohom-indZ2 |Hom(C2r +1 , G )| + 1 ≤ cohom-indZ2 |Hom(K2 , G )|.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
15 / 20
The easy part
The composition map
Hom(K2 , C2r +1 ) × Hom(C2r +1 , G ) −−→ Hom(K2 , G )
yields
S1b ×Z2 |Hom(C2r +1 , G )| −−→Z2 |Hom(K2 , G )|.
Direct consequence:
cohom-indZ2 |Hom(C2r +1 , G )| + 1 ≤ cohom-indZ2 |Hom(K2 , G )|.
This already proves the Lovász Conjecture.
Živaljević, R. T.
math.CO/0506075.
Parallel transport of Hom-complexes and the Lovász conjecture, 2005.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
15 / 20
The easy part
The composition map
Hom(K2 , C2r +1 ) × Hom(C2r +1 , G ) −−→ Hom(K2 , G )
yields
S1b ×Z2 |Hom(C2r +1 , G )| −−→Z2 |Hom(K2 , G )|.
Direct consequence:
cohom-indZ2 |Hom(C2r +1 , G )| + 1 ≤ cohom-indZ2 |Hom(K2 , G )|.
This already proves the Lovász Conjecture.
The adjoint maps
|Hom(C2r +1 , G )| −−→Z2 MapZ2 (S1b , |Hom(K2 , G )|)
can be fitted together to
colim|Hom(C2r +1 , G )| −−→Z2 MapZ2 (S1b , |Hom(K2 , G )|).
r
Živaljević, R. T.
math.CO/0506075.
Parallel transport of Hom-complexes and the Lovász conjecture, 2005.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
15 / 20
A further hint at the relationship between
Hom(K2 , G ) and Hom(C2r +1 , G ).
Proposition
MapZ2 (S1b , |Hom(K2 , G )|) 6= Ø
⇐⇒ |Hom(C2r +1 , G )| =
6 Ø for r large enough.
Proof.
4
5
3
6
2
1
7
8
Lovász, L. Kneser’s conjecture, chromatic number and homotopy.
Theory, Ser. A, 25:319–324, 1978.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
J. Combinatorial
Stockholm ’06
16 / 20
Outline
1
Main result
2
Examples and Consequences
Cycles in complete and cyclic graphs
Graph colouring obstructions
3
Bits of the proof
4
Preview of further results
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
17 / 20
Generalizations
joint with Babson & Dochtermann
Definition
We define graphs Tk,r for k, r ≥ 1.
a
T1,r = C2r +1
e
d
c
b
a
i
i
h
h
g
g
f
f
a
b
c
d
e
a
T2,2
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
18 / 20
Generalizations
joint with Babson & Dochtermann
Definition
We define graphs Tk,r for k, r ≥ 1.
(T1,r = C2r +1 , . . . )
Theorem
cohom-indZ2 |Hom(Tk,r , G )| + k ≤ cohom-indZ2 |Hom(K2 , G )|
lim coindZ2 |Hom(Tk,r , G )| + k ≥ coindZ2 |Hom(K2 , G )|
r →∞
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
19 / 20
Generalizations
joint with Babson & Dochtermann
Definition
We define graphs Tk,r for k, r ≥ 1.
(T1,r = C2r +1 , . . . )
Theorem
cohom-indZ2 |Hom(Tk,r , G )| + k ≤ cohom-indZ2 |Hom(K2 , G )|
lim coindZ2 |Hom(Tk,r , G )| + k ≥ coindZ2 |Hom(K2 , G )|
r →∞
Corollary
coindZ2 |Hom(K2 , G )| ≤ max {k : Ex. r ≥ 1 and Tk,r → G }
≤ cohom-indZ2 |Hom(K2 , G )| ≤ χ(G ) − 2.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
19 / 20
Generalizations
joint with Babson & Dochtermann
Definition
We define graphs Tk,r for k, r ≥ 1.
(T1,r = C2r +1 , . . . )
Theorem
cohom-indZ2 |Hom(Tk,r , G )| + k ≤ cohom-indZ2 |Hom(K2 , G )|
lim coindZ2 |Hom(Tk,r , G )| + k ≥ coindZ2 |Hom(K2 , G )|
r →∞
Corollary
coindZ2 |Hom(K2 , G )| ≤ max {k : Ex. r ≥ 1 and Tk,r → G }
≤ cohom-indZ2 |Hom(K2 , G )| ≤ χ(G ) − 2.
χ(Tk,r ) ≥ k + 2, χ(Tk,r ) = k + 2, if r is large enough.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
19 / 20
Generalizations
joint with Babson & Dochtermann
Definition
We define graphs Tk,r for k, r ≥ 1.
(T1,r = C2r +1 , . . . )
Theorem
cohom-indZ2 |Hom(Tk,r , G )| + k ≤ cohom-indZ2 |Hom(K2 , G )|
lim coindZ2 |Hom(Tk,r , G )| + k ≥ coindZ2 |Hom(K2 , G )|
r →∞
Corollary
coindZ2 |Hom(K2 , G )| ≤ max {k : Ex. r ≥ 1 and Tk,r → G }
≤ cohom-indZ2 |Hom(K2 , G )| ≤ χ(G ) − 2.
χ(Tk,r ) ≥ k + 2, χ(Tk,r ) = k + 2, if r is large enough.
If e.g. K is a Kneser graph with χ(K ) = k + 2, then
coindZ2 |Hom(K2 , K )| = k and hence there exist r and Tk,r → K .
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
19 / 20
Summary
In the limit, the Z2 -homotopy type of Hom(C2r +1 , G ) is determined
by the Z2 -homotopy type of Hom(K2 , G ).
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
20 / 20
Summary
In the limit, the Z2 -homotopy type of Hom(C2r +1 , G ) is determined
by the Z2 -homotopy type of Hom(K2 , G ).
The bounds on χ(G ) obtained from Hom(K2 , G ) and Hom(C2r +1 , G )
for large r are essentially the same.
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
20 / 20
Summary
In the limit, the Z2 -homotopy type of Hom(C2r +1 , G ) is determined
by the Z2 -homotopy type of Hom(K2 , G ).
The bounds on χ(G ) obtained from Hom(K2 , G ) and Hom(C2r +1 , G )
for large r are essentially the same.
coindZ2 Hom(K2 , G ) can be described combinatorially via the
existence of graph homomorphisms to G .
Carsten Schultz (TU Berlin)
The Lovász Conjecture and extensions
Stockholm ’06
20 / 20