Hom complexes and hypergraph colorings Daisuke Kishimoto - PowerPoint PPT Presentation

Hom complexes and hypergraph colorings Daisuke Kishimoto Department of Mathematics Kyoto University in collaboration with K. Iriye 8 December 2011; EACAT4

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). 2-uniform hypergraph 3-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). 2-uniform hypergraph 3-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). 2-uniform hypergraph 3-uniform hypergraph 4-uniform hypergraph Definition: Let G,H be r-uniform hypergraphs. A homomorphism f:G → H is a map f:V(G) → V(H) whenever {v 1 ,...,v r } ∈ E(G), {f(v 1 ),...,f(v r )} ∈ E(H). get a category of r-uniform hypergraphs and homomorphisms.

Hypergraph colorings

Hypergraph colorings Definition: An n-coloring of G is a map c:V(G) → [n]={1,...,n} whenever {v 1 ,...,v r } ∈ E(G), {f(v 1 ),...,f(v r )} ⊂ [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 {v 1 ,...,v r } ∈ E(G), {f(v 1 ),...,f(v r )} ⊂ [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 {v 1 ,...,v r } ∈ E(G), {f(v 1 ),...,f(v r )} ⊂ [n] is not a one point set. 3-coloring of a 2-uniform 2-coloring of a 3-uniform hypergraph hypergraph

Hypergraph colorings Definition: An n-coloring of G is a map c:V(G) → [n]={1,...,n} whenever {v 1 ,...,v r } ∈ E(G), {f(v 1 ),...,f(v r )} ⊂ [n] is not a one point set. 3-coloring of a 2-uniform 2-coloring of a 3-uniform 4-coloring of a 4-uniform hypergraph hypergraph hypergraph

Hypergraph colorings Definition: An n-coloring of G is a map c:V(G) → [n]={1,...,n} whenever {v 1 ,...,v r } ∈ E(G), {f(v 1 ),...,f(v r )} ⊂ [n] is not a one point set. 3-coloring of a 2-uniform 2-coloring of a 3-uniform 4-coloring of a 4-uniform hypergraph hypergraph 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 | ∀ (v 1 ,...,v r ) ∈ ( π 1 (F),..., π r (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 | ∀ (v 1 ,...,v r ) ∈ ( π 1 (F),..., π r (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 | ∀ (v 1 ,...,v r ) ∈ ( π 1 (F),..., π r (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. 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.

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

Hom complexes of graphs Assume that graphs are 2-uniform hypergraphs. Let K n be the graph such that V(K n )=[n] and E(K n ) is the set of all 2- subsets of [n]. an n-coloring of a graph G ↔ a homomorphism G → K n 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 Δ T S Definition: Let G,H be graphs. The Hom complex Hom(G,H) is the V(H) maximum subcomplex of Π Δ whose vertices are homomorphisms V(G) G → H.

Result of Babson and Kozlov

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

Result of Babson and Kozlov Hom complexes yield a functor Graphs op × Graphs → Spaces . In particular, the Z /2-action on K 2 by flipping induces the Z /2-action on Hom(K 2 ,G). Lemma [Babson-Kozlov ‘06]: The Z /2-action on Hom(K 2 ,G) is free. Proposition [Babson-Kozlov ‘06]: Hom(K 2 ,K n ) ≃ S n-2 .

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

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

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 of 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 of 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 Example: 0 0 1 3 0 0 2 set S 6-multisubset of S 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 of 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 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 of 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. 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. (r) (r) Define the r-graph K n as V(K n )= [n] and E(G) to be the set of all non- trivial r-multisubsets of [n]. (r) an n-coloring of an r-graph G ↔ a homomorphism G → K n

Hom complexes of r-graphs