Hom complexes and hypergraph colorings Daisuke Kishimoto - - PowerPoint PPT Presentation

Hom complexes and hypergraph colorings

Daisuke Kishimoto

Department of Mathematics Kyoto University 8 December 2011; EACAT4 in collaboration with K. Iriye

r-uniform hypergraphs

r-uniform hypergraphs

Definition: An r-uniform hypergraph G consists of the vertex set V(G) (a finite set) and the edge set E(G) which is a collection of r-subsets of V(G).

r-uniform hypergraphs

Definition: An r-uniform hypergraph G consists of the vertex set V(G) (a finite set) and the edge set E(G) which is a collection of r-subsets of V(G).

2-uniform hypergraph

r-uniform hypergraphs

Definition: An r-uniform hypergraph G consists of the vertex set V(G) (a finite set) and the edge set E(G) which is a collection of r-subsets of V(G).

3-uniform hypergraph 2-uniform hypergraph

r-uniform hypergraphs

Definition: An r-uniform hypergraph G consists of the vertex set V(G) (a finite set) and the edge set E(G) which is a collection of r-subsets of V(G).

3-uniform hypergraph 2-uniform hypergraph 4-uniform hypergraph

r-uniform hypergraphs

Definition: An r-uniform hypergraph G consists of the vertex set V(G) (a finite set) and the edge set E(G) which is a collection of r-subsets of V(G).

3-uniform hypergraph

get a category of r-uniform hypergraphs and homomorphisms. Definition: Let G,H be r-uniform hypergraphs. A homomorphism f:G→H is a map f:V(G)→V(H) whenever {v1,...,vr}∈E(G), {f(v1),...,f(vr)}∈E(H).

2-uniform hypergraph 4-uniform hypergraph

Hypergraph colorings

Hypergraph colorings

Definition: An n-coloring of G is a map c:V(G)→[n]={1,...,n} whenever {v1,...,vr}∈E(G), {f(v1),...,f(vr)}⊂[n] is not a one point set.

Hypergraph colorings

Definition: An n-coloring of G is a map c:V(G)→[n]={1,...,n} whenever {v1,...,vr}∈E(G), {f(v1),...,f(vr)}⊂[n] is not a one point set.

3-coloring of a 2-uniform hypergraph

Hypergraph colorings

Definition: An n-coloring of G is a map c:V(G)→[n]={1,...,n} whenever {v1,...,vr}∈E(G), {f(v1),...,f(vr)}⊂[n] is not a one point set.

2-coloring of a 3-uniform hypergraph 3-coloring of a 2-uniform hypergraph

Hypergraph colorings

Definition: An n-coloring of G is a map c:V(G)→[n]={1,...,n} whenever {v1,...,vr}∈E(G), {f(v1),...,f(vr)}⊂[n] is not a one point set.

2-coloring of a 3-uniform hypergraph 3-coloring of a 2-uniform hypergraph 4-coloring of a 4-uniform hypergraph

Hypergraph colorings

Definition: An n-coloring of G is a map c:V(G)→[n]={1,...,n} whenever {v1,...,vr}∈E(G), {f(v1),...,f(vr)}⊂[n] is not a one point set.

2-coloring of a 3-uniform hypergraph 3-coloring of a 2-uniform hypergraph 4-coloring of a 4-uniform hypergraph

Definition: The minimum n such that ∃n-coloring of G is called the chromatic number of G and denoted by χ(G). Example: The chromatic numbers of the above hypergraphs are 2.

Result of Alon, Frankl and Lovasz

́

Result of Alon, Frankl and Lovasz

Definition: For an r-uniform hypergraph G, define a simplicial complex B(G)={F⊂V(G)r | ∀(v1,...,vr)∈(π1(F),...,πr(F)) belongs to E(G)}.

́

Result of Alon, Frankl and Lovasz

Definition: For an r-uniform hypergraph G, define a simplicial complex B(G)={F⊂V(G)r | ∀(v1,...,vr)∈(π1(F),...,πr(F)) belongs to E(G)}.

́

Theorem [Alon, Frankl and Lovasz ‘86]: If r is a prime and B(G) is n- connected, χ(G)≧(n+2)/(r-1)+1. ́

Result of Alon, Frankl and Lovasz

Definition: For an r-uniform hypergraph G, define a simplicial complex B(G)={F⊂V(G)r | ∀(v1,...,vr)∈(π1(F),...,πr(F)) belongs to E(G)}.

́

Proof: Z/r acts freely on B(G) by rotating entries of V(G)r. Define a Z/r- action on R(r-1)(n-1)-0 satisfying:

• 1. It is free if r is a prime.
• 2. If G is n-colorable, there is a Z/r-map B(G)→ R(r-1)(n-1)-0.

Then the result follows from the Borsuk-Ulam theorem. Theorem [Alon, Frankl and Lovasz ‘86]: If r is a prime and B(G) is n- connected, χ(G)≧(n+2)/(r-1)+1. ́

Motivation

Motivation

Unsatisfactory points of the above result:

• 1. What’s the meaning of B(G)?
• 2. How does the Z/r-action on R(r-1)(n-1)-0 comes up?

Motivation

Unsatisfactory points of the above result:

• 1. What’s the meaning of B(G)?
• 2. How does the Z/r-action on R(r-1)(n-1)-0 comes up?

Go back to graph colorings. Then the meaning of Hom complexes is clear (mapping spaces) and they work successfully for graph colorings. Want to generalize Hom complexes of graphs to r-uniform hypergraphs and get a good understanding.

Hom complexes of graphs

Hom complexes of graphs

Assume that graphs are 2-uniform hypergraphs.

Hom complexes of graphs

Let Kn be the graph such that V(Kn)=[n] and E(Kn) is the set of all 2- subsets of [n]. an n-coloring of a graph G ↔ a homomorphism G→Kn Then we consider “the space of graph homomorphisms“. ...but what should it be? Assume that graphs are 2-uniform hypergraphs.

Hom complexes of graphs

Let Kn be the graph such that V(Kn)=[n] and E(Kn) is the set of all 2- subsets of [n]. an n-coloring of a graph G ↔ a homomorphism G→Kn Then we consider “the space of graph homomorphisms“. ...but what should it be? Let S,T be finite sets and let Δ be the simplex whose vertex set is T.

T

a map from S to T ↔ a vertex of Π ΔT

S

Definition: Let G,H be graphs. The Hom complex Hom(G,H) is the maximum subcomplex of Π Δ whose vertices are homomorphisms G→H.

V(G) V(H)

Assume that graphs are 2-uniform hypergraphs.

Result of Babson and Kozlov

Result of Babson and Kozlov

Hom complexes yield a functor Graphsop × Graphs → Spaces. In particular, the Z/2-action on K2 by flipping induces the Z/2-action on Hom(K2,G). Lemma [Babson-Kozlov ‘06]: The Z/2-action on Hom(K2,G) is free.

Result of Babson and Kozlov

Hom complexes yield a functor Graphsop × Graphs → Spaces. In particular, the Z/2-action on K2 by flipping induces the Z/2-action on Hom(K2,G). Lemma [Babson-Kozlov ‘06]: The Z/2-action on Hom(K2,G) is free. Proposition [Babson-Kozlov ‘06]: Hom(K2,Kn)≃Sn-2.

Result of Babson and Kozlov

Hom complexes yield a functor Graphsop × Graphs → Spaces. In particular, the Z/2-action on K2 by flipping induces the Z/2-action on Hom(K2,G). Lemma [Babson-Kozlov ‘06]: The Z/2-action on Hom(K2,G) is free. Proposition [Babson-Kozlov ‘06]: Hom(K2,Kn)≃Sn-2. Theorem [Babson-Kozlov ‘06]: If Hom(K2,G) is n-connected, then χ(G)≧n+3.

Result of Babson and Kozlov

Hom complexes yield a functor Graphsop × Graphs → Spaces. In particular, the Z/2-action on K2 by flipping induces the Z/2-action on Hom(K2,G). Lemma [Babson-Kozlov ‘06]: The Z/2-action on Hom(K2,G) is free. Proposition [Babson-Kozlov ‘06]: Hom(K2,Kn)≃Sn-2. Proof: If G is n-colorable, there is a Z/2-map Hom(K2,G)→ Hom(K2,Kn)≃Sn-2. Then the result follows from the Borsuk-Ulam theorem. Theorem [Babson-Kozlov ‘06]: If Hom(K2,G) is n-connected, then χ(G)≧n+3.

Generalizing r-uniform hypergraphs

Generalizing r-uniform hypergraphs

Example: A 2-coloring of a 3-uniform hypergraph which is not realized by any homomorphism. Hom complexes never work for colorings if we work in the category

• f r-uniform hypergraphs and homomorphisms between them.

Generalizing r-uniform hypergraphs

Example: A 2-coloring of a 3-uniform hypergraph which is not realized by any homomorphism. Hom complexes never work for colorings if we work in the category

• f r-uniform hypergraphs and homomorphisms between them.

Generalize r-uniform hypergraphs so that colorings can be realized as homomorphisms. ...but how? Observation: Regard the above colored 3-uniform hypergraph as: 1 2 Consider “weighted” sets.

r-multisubset

r-multisubset

Definition: An r-multisubset of a finite set S is a map w:S→Z≧0 satisfying Σ w(s)=r.

s∈S

r-multisubset

Definition: An r-multisubset of a finite set S is a map w:S→Z≧0 satisfying Σ w(s)=r.

s∈S

set S 6-multisubset of S 1 2 3

Example: Example: An r-subset T of S is an r-multisubset by w:S→Z≧0 such that w(s)=1and 0 according as s∈T and s∉T. Example: An r-multisubset w:S→Z≧0 is called trivial if for some s∈S, w(s)=r (and then w(t)=0 for t≠s).

r-graphs

r-graphs

Definition: An r-graph G consists of the vertex set V(G) (a finite set) and the edge set E(G) which is a collection of non-trivial r-multisubsets

• f V(G).

Definition: Let G,H be r-graphs. A homomorphism f:G→H is a map f:V(G)→V(H) whenever e∈E(G), f(e)∈E(H). get a category of r-graphs and homomorphisms.

r-graphs

Definition: An n-coloring of an r-graph G is a map c:V(G)→[n] whenever e∈V(G), c(e) is not trivial. Define the r-graph Kn as V(Kn )= [n] and E(G) to be the set of all non- trivial r-multisubsets of [n].

(r) (r)

an n-coloring of an r-graph G ↔ a homomorphism G→Kn

(r)

Definition: An r-graph G consists of the vertex set V(G) (a finite set) and the edge set E(G) which is a collection of non-trivial r-multisubsets

• f V(G).

Definition: Let G,H be r-graphs. A homomorphism f:G→H is a map f:V(G)→V(H) whenever e∈E(G), f(e)∈E(H). get a category of r-graphs and homomorphisms.

Hom complexes of r-graphs

Hom complexes of r-graphs

Definition: Let G,H be r-graphs. The Hom complex Hom(G,H) is the maximum subcomplex of Π Δ whose vertices are homomorphisms G→H.

V(H) V(G)

yielding a functor r-Graphsop × r-Graphs → Spaces.

Hom complexes of r-graphs

Definition: Let G,H be r-graphs. The Hom complex Hom(G,H) is the maximum subcomplex of Π Δ whose vertices are homomorphisms G→H.

V(H) V(G)

yielding a functor r-Graphsop × r-Graphs → Spaces. Define Kn as V(Kn )=[n] and E(Kn ) is the set of all r-subsets of [n].

(r) (r)

Z/r acts on Kr by rotating vertices, inducing the Z/r-action on Hom(Kr ,G).

(r) (r) (r)

Proposition: If r is a prime, the Z/r-action on Hom(Kr ,G) is free.

(r)

The homotopy type of Hom(Km ,Kn )

(r) (r)

The homotopy type of Hom(Km ,Kn )

For (t1,...,tn)∈[m]n, let Δ(t1,...,tn) be the subcomplex of Π Δ whose faces are (F1,...,Fm)⊂ Π Δ such that 1≦♯{ j | i∈Fj }≦ti for each i∈[n].

[n] m m [n]

Then Δ(r-1,...,r-1)= Hom(Km ,Kn ).

(r) (r)

(r) (r)

The homotopy type of Hom(Km ,Kn )

For (t1,...,tn)∈[m]n, let Δ(t1,...,tn) be the subcomplex of Π Δ whose faces are (F1,...,Fm)⊂ Π Δ such that 1≦♯{ j | i∈Fj }≦ti for each i∈[n].

[n] m m [n]

Then Δ(r-1,...,r-1)= Hom(Km ,Kn ).

(r) (r)

Proposition: Δ(t1,...,tn) has the homotopy type of a wedge of ( Σ ti -m)- dimensional spheres.

i=1 n

Corollary: Hom(Km ,Kn ) has the homotopy type of a wedge of ((r-1)n- m)-dimensional spheres.

(r) (r)

(r) (r)

Γ-index

Γ-index

It is useful to introduce the Γ-index to make things clear. Definition: Let Γ be a finite group and let X be a free Γ-space. The Γ- index of Γ, denoted by indΓX is the minimum n such that there is a free Γ-complex which is (n-1)-connected and n-dimensional.

Γ-index

Proposition: Let Γ be a finite group and let X,Y be free Γ-spaces.

• 1. If there is a Γ-map X→Y, indΓX≦indΓY.
• 2. indΓX≧connectivity of X+1 (Borsuk-Ulam theorem)
• 3. The join X＊Y is a free Γ-space with indΓ(X＊Y)≦indΓX+indΓY+1.

It is useful to introduce the Γ-index to make things clear. Definition: Let Γ be a finite group and let X be a free Γ-space. The Γ- index of Γ, denoted by indΓX is the minimum n such that there is a free Γ-complex which is (n-1)-connected and n-dimensional.

Lower bounds for chromatic numbers

Lower bounds for chromatic numbers

Theorem: Let G be an r-graph. If r is a prime, χ(G) ≧ (indZ/rHom(Kr ,G)+1)/(r-1)+1.

(r)

Lower bounds for chromatic numbers

Theorem: Let G be an r-graph. If r is a prime, χ(G) ≧ (indZ/rHom(Kr ,G)+1)/(r-1)+1.

(r)

Proof: If G is n-colorable, there is a Z/r-map Hom(Kr ,G)→Hom(Kr ,Kn )≃∨S(r-1)n-r. Then indZ/rHom(Kr ,G)≦(r-1)n-r=(r-1)(n-1)-1, completing the proof.

(r) (r) (r) (r)

Lower bounds for chromatic numbers

Theorem: Let G be an r-graph. If r is a prime, χ(G) ≧ (indZ/rHom(Kr ,G)+1)/(r-1)+1.

(r)

Proof: If G is n-colorable, there is a Z/r-map Hom(Kr ,G)→Hom(Kr ,Kn )≃∨S(r-1)n-r. Then indZ/rHom(Kr ,G)≦(r-1)n-r=(r-1)(n-1)-1, completing the proof.

(r) (r) (r) (r)

Corollary: If r is a prime and Hom(Kr ,G) is n-connected, then χ(G) ≧ (n+2)(r-1)+1.

(r)

Lower bounds for chromatic numbers

Theorem: Let G be an r-graph. If r is a prime, χ(G) ≧ (indZ/rHom(Kr ,G)+1)/(r-1)+1.

(r)

Proof: If G is n-colorable, there is a Z/r-map Hom(Kr ,G)→Hom(Kr ,Kn )≃∨S(r-1)n-r. Then indZ/rHom(Kr ,G)≦(r-1)n-r=(r-1)(n-1)-1, completing the proof.

(r) (r) (r) (r)

Corollary: If r is a prime and Hom(Kr ,G) is n-connected, then χ(G) ≧ (n+2)(r-1)+1.

(r)

Proof: By the Borsuk-Ulam theorem, indZ/rHom(Kr ,G)+1≧ n.

(r)

B(G) and Hom complexes

B(G) and Hom complexes

Proposition: There is a Z/r-map B(G)→Hom(Kr ,G) which is a homotopy equivalence. In particular, if r is a prime, it is a Z/r-homotopy equivalence.

(r)

Remark: Recently, Thansri showed that this map is a Z/r-equivariant simplicial collapse.

B(G) and Hom complexes

Proposition: The Z/r-action on Hom(Kr ,Kn )⊂ Π Δ ⊂Rr(n-1) - Rn-1 extends to the Alon-Frankl-Lovasz Z/r-action on R(r-1)(n-1)-0.

(r) (r) r [n]

Proposition: There is a Z/r-map B(G)→Hom(Kr ,G) which is a homotopy equivalence. In particular, if r is a prime, it is a Z/r-homotopy equivalence.

(r)

Remark: Recently, Thansri showed that this map is a Z/r-equivariant simplicial collapse.

Other results

Other results

Lange introduced the join version of B(G) and obtained a result analogous to Alon-Frankl-Lovasz. Using Hom+ complexes (join version

• f Hom complexes), we can interpret his construction. In particular, we

have: ́

Other results

Lange introduced the join version of B(G) and obtained a result analogous to Alon-Frankl-Lovasz. Using Hom+ complexes (join version

• f Hom complexes), we can interpret his construction. In particular, we

have: ́ Proposition: There is a Z/r-map from Lange’s complex to ∂Δ ＊Hom(Kr ,G) which is a homotopy equivalence. Then if p is a prime, it is a Z/r-homotopy equivalence.

(r) [r]

Other results

Proposition: There is a hierarchy of lower bounds for chromaric numbers: AFL-1 ≦ Lange ≦ AFL (≦ χ(G)). Lange introduced the join version of B(G) and obtained a result analogous to Alon-Frankl-Lovasz. Using Hom+ complexes (join version

• f Hom complexes), we can interpret his construction. In particular, we

have: ́ Proposition: There is a Z/r-map from Lange’s complex to ∂Δ ＊Hom(Kr ,G) which is a homotopy equivalence. Then if p is a prime, it is a Z/r-homotopy equivalence.

(r) [r]

Other results

Proposition: There is a hierarchy of lower bounds for chromaric numbers: AFL-1 ≦ Lange ≦ AFL (≦ χ(G)). Lange introduced the join version of B(G) and obtained a result analogous to Alon-Frankl-Lovasz. Using Hom+ complexes (join version

• f Hom complexes), we can interpret his construction. In particular, we

have: ́ Proposition: There is a Z/r-map from Lange’s complex to ∂Δ ＊Hom(Kr ,G) which is a homotopy equivalence. Then if p is a prime, it is a Z/r-homotopy equivalence.

(r) [r]

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