Complexes of Graph Homomorphisms D.N. Kozlov: Chromatic numbers, - - PDF document

Complexes of Graph Homomorphisms

D.N. Kozlov: Chromatic numbers, morphism complexes, and Stiefel-Whitney characteristic classes; IAS/Park City Mathematics Series 14,

• Amer. Math. Soc., Providence, RI;

Institute for Advanced Study, Princeton, NJ.

• E. Babson, D.N. Kozlov: Proof of the Lov´

asz Conjecture; Annals of Mathematics, to appear. D.N. Kozlov: Cohomology of colorings of cycles; arXiv:math.AT/0507117

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Fragestellung: How many colors does one need to color the vertices of a given graph G, so that if two vertices are connected by an edge, then they get different colors? The minimal number of colors is called chromatic number and is denoted by χ(G). In general, it is very difficult to compute χ(G) or even to find bounds. It is even NP-hard to decide whether χ(G) = 3.

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Ansatz: Graph G − → Topological space X(G) ↓ Combinatorial ← − Topological properties of G properties of X(G) The topological spaces occurring in this context are finite regular CW complexes, where the cells are products of simplices. At our disposal we have theorems expressing topological obstruc- tions, such as the Borsuk-Ulam theorem.

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Historic Interlude.

Definition. Let G be a graph. The neighborhood complex N (G) is a simplicial complex defined as follows:

• vertices of N(G) are the vertices of G;
• a set of vertices A forms a simplex if and only if all vertices in

A have a common neighbor. In particular, every vertex v in G defines a maximal simplex con- sisting of all neighbors of v. Examples.

1 2 3 4 1 2 3 4

K4 N (K4)

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2,3 2,3 3,4 4,5 1,3 2,4 1,5 2,5 3,5 1,4 1,2 a b c e f d d f c e a b

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Definition. A topological space C is called k-connected, if every continuous map φ : Sm → C can be extended to a map φ : Bm+1 → C, for any −1 ≤ m ≤ k. Equivalently: homotopy groups up to dimension k are trivial. Theorem (Lov´ asz, 1978). Let G be a graph and k ∈ Z, k ≥ −1. Then N(G) is k-connected = ⇒ χ(G) ≥ k + 3 . Using this theorem Lov´ asz has proved the Kneser Conjecture.

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Equivalently, Lov´ asz’ theorem can be formulated as follows: Hom (K2, G) is k-connected = ⇒ χ(G) ≥ k + 3 . We shall define Hom (−, −) shortly. Lov´ asz Conjecture. Let G be a graph and r, k ∈ Z, r ≥ 1, k ≥ −1. Then Hom (C2r+1, G) is k-connected = ⇒ χ(G) ≥ k + 4 . Theorem (Babson & K., 2003). (a) The Lov´ asz Conjecture is true. (b) Let m, k ∈ Z, m ≥ 1, k ≥ −1. Then Hom (Km, G) is k-connected = ⇒ χ(G) ≥ k + m + 1 .

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Definition. Let T, G be two graphs. A graph homomorphism from T to G is a map φ : V (T) → V (G), such that for every edge (x, y) in T the image (φ(x), φ(y)) is an edge in G. Observation: G is n-colorable ⇔ ∃ φ : G → Kn. A composition of two homomorphisms φ1 : G1 → G2 and φ2 : G2 → G3 is again a homomorphism φ2 ◦ φ1 : G1 → G3. This gives Graphs - the category which has graphs as objects and graph homomorphisms as morphisms.

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Geometry of graph colorings: idea.

A cell in Hom (T, Kn) is a collection of color lists, one for each vertex

• f T, such that an arbitrary choice of colors, one from each list, is

a valid coloring of T. Now replace Kn with an arbitrary graph G.

• The vertices of G are the colors.
• A homomorphism T −

→ G is thought of as a valid coloring. Example.

Hom( , ) =

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Definition. Let T and G be two graphs. The Hom -complex Hom (T, G) is the subcomplex of

x∈V (T) ∆V (G) defined by the following condition:

the cell σ =

x∈V (T) σx is in Hom (T, G) if and only if whenever two

vertices x, y ∈ V (T) are connected by an edge, the pair (σx, σy) is a complete bipartite subgraph of G. Note that

• the cells of Hom (T, G) are indexed by all set maps

η : V (T) → 2V (G) \{∅}, such that for any (x, y) ∈ E(T), and any ˜ x ∈ η(x), ˜ y ∈ η(y), we have (˜ x, ˜ y) ∈ E(G);

• the cells in the closure of each cell η are indexed by all maps

˜ η : V (T) → 2V (G) \ {∅}, such that ˜ η(v) ⊆ η(v), for all v ∈ V (T).

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3 12 3 12 3 12 23 1 23 1 23 1 1 2 1 2 1 2 1 23 1 2 13 2 1 2 13 2 1 23 1 23 1 23 1 23 12 3 12 3 12 3 2 13 2 13 2 13 1 2 3 1 2 3 13 2 13 2 13 2

3 12 3 12 3 12 23 1 23 1 23 1 1 2 1 2 1 2 1 23 1 2 13 2 1 2 13 2 1 23 1 23 1 23 1 23 12 3 12 3 12 3 2 13 2 13 2 13 1 2 3 1 2 3 13 2 13 2 13 2

Hom (C6, K3)

2 1 23 2 2 2 2

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2 13 1 13 13 2 1 2 2 13 23 1 1 1 1 23 23 2 1 1 23 23

1 2 1 13 1 2

2 1 23 2 2 2 2

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2 13 1 13 13 2 1 2 2 13 23 1 1 1 1 23 23 2 1 1 23 23

1 2 1 13 1 2

v = 11

Properties of the Hom -complexes: (1) Cells in Hom (T, G) are products of simplices: this is a prodsimplicial complex. More precisely, every cell η is a product of |V (T)| simplices of dimension |η(x)| − 1 for x ∈ V (T). (2) Bd Hom (K2, G) and N(G) have the same simple homotopy type. (3) Hom (T, −) is a covariant and Hom (−, G) is a contravariant functor from Graphs to Top. → If φ : G → G′ is a graph homomorphism, then, for an arbitrary graph H, we shall denote the induced topological maps by φH : Hom (H, G) → Hom (H, G′) and φH : Hom (G′, H) → Hom (G, H). → As a consequence of (3), the group Aut(T) × Aut(G) acts on Hom (T, G). → When G has no loops and φ ∈ Aut(T) flips an edge, the induced map φG Hom (T, G) has no fixed points.

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Definition. A CW complex X is called Z2-space, if Z2 acts freely on X. In this case, there exists a continuous map ψ : X → S∞. The induced quotient map φ : X/Z2 → RP∞ is up to homotopy inde- pendent of the choice of ψ. It induces an algebra map φ∗ : H∗(RP∞; Z2) → H∗(X/Z2; Z2), which is independent of the choice of ψ. Let z be the generator of H1(RP∞; Z2). Then w1(X):= φ∗(z) is called the Stiefel-Whitney class.

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Stiefel-Whitney classes are functorial: if ψ : X → Y is a Z2-map, and ψ : X/Z2 → Y/Z2 is the induced map between the quotient spaces, then

• ψ∗(w1(Y )) = w1(X) .

An application of Stiefel-Whitney classes: Borsuk-Ulam Theorem. There is no continuous map Sk+1 − → Sk, which commutes with the antipodal maps on Sk+1 and Sk.

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Lov´ asz Conjecture: Hom (C2r+1, G) is k-connected = ⇒ χ(G) ≥ k + 4 . k = −1 : Hom (C2r+1, G) is nonempty = ⇒ χ(G) ≥ 3 . k = 0 : Hom (C2r+1, G) is connected = ⇒ χ(G) ≥ 4 . Proof for k = 0. Assume χ(G) ≤ 3, then there exists a graph homomorphism φ : G → K3. It induces a Z2-map φC2r+1 : Hom (C2r+1, G) → Hom (C2r+1, K3). A direct analysis shows that the connected components of the com- plex Hom (C2r+1, K3) can be indexed with the signed number of times C2r+1 winds around K3. This number α is odd: α = ±1, ±3, . . . , ±(2s + 1), and s ≥ 0. The Z2-action on Hom (C2r+1, K3) swaps the connected components by changing the sign of the winding number. This contradicts the fact that Hom (C2r+1, G) is connected.

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Scheme of the Proof (Babson & K., 2003):

• Since Hom (C2r+1, G) is (n − 1)-connected there exists a Z2-map

f : Sn

a → Hom (C2r+1, G).

• Assume χ(G) ≤ n + 2, then ∃ φ : G → Kn+2.

This induces a Z2-map φC2r+1 : Hom (C2r+1, G) → Hom (C2r+1, Kn+2).

• Since φC2r+1 ◦ f : Sn

a → Hom (C2r+1, Kn+2) is a Z2-map and

wn

1(Sn a) = 0, we conclude that

wn

1(Hom (C2r+1, Kn+2)) = 0.

• On the other hand, for odd n, spectral sequence computations

yield wn

1(Hom (C2r+1, Kn+2)) = 0 =

⇒ a contradiction.

• For even n the obstructions are found simply by computing

H∗(Hom (C2r+1, Kn); Z).

• Further work on L´
• vasz Conjecture and our (BK) conjecture that

wn

1(Hom (C2r+1, Kn+2)) = 0 also for even n by ˇ

Zivaljevi´ c, Schultz.

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Facts about Hom ’s I.

• Theorem. (Babson & K., 2003)

Let m, n ∈ Z, n ≥ m ≥ 2. Then Hom (Km, Kn) is homotopy equivalent to a wedge of (n − m) - dimensional spheres. Note that Hom (K2, Kn) can be realized as a boundary of an (n−2) - dimensional polytope, in particular it is homeomorphic to Sn−2. Example.

Hom (K2, K4) =

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Facts about Hom ’s II. Replacing G with G − v is called a fold. Some special cases:

• A tree folds onto any of its edges.
• Let F be a forest, then ¯

F folds onto Km, where m is the maximal cardinality of an independent set in F.

• Theorem. (K., 2004)

Let G and H be two graphs, and let u, v be vertices of T, such that N(v) ⊆ N(u). Then Bd Hom (G, H) collapses onto Bd Hom (G−v, H), whereas Hom (H, G) collapses onto Hom (H, G − v).

Hom (L3, K3) Hom (K2, K3) X 18

Facts about Hom ’s III.

• Theorem. (ˇ

Cuki´ c & K., 2004) Let G be a graph of maximal valency d, then the complex Hom (G, Kn) is at least (n − d − 2)-connected. This was conjectured by Babson & K., short proof: Engstr¨

• m,

2005.

• Theorem. (ˇ

Cuki´ c & K., 2004) Every connected component of Hom (Cm, Cn) is either a point or is homotopicaly equivalent to S1.

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Facts about Hom ’s VI. Universality Theorem. (Csorba, ˇ Zivaljevi´ c, 2004) For each finite, free Z2-complex X, there exists a graph G, such that Hom (K2, G) is Z2-homotopy equivalent to X.

• Conjecture. (Csorba, 2004)

Hom (C5, Kn) is homeomorphic to the Stiefel manifold V2(Rn−1), for all n ≥ 1. For example, Hom (C5, K4) ∼ = RP3.

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Cohomology of complexes of cycles I. Hom +-construction. Similar to Hom , empty coloring lists are allowed.

• Hom +(T, G) is a simplicial complex.
• Hom +(T, Kn) is isomorphic to the n-fold join of the indepen-

dence complex of T.

• There exists a simplicial map

supp : Hom +(T, G) → ∆|V (T)| .

• Hom (T, G) = supp−1(ρ0), where ρ0 is the barycenter of the

simplex ∆|V (T)|.

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Cohomology of complexes of cycles II.

• The spectral sequence: the filtration on Hom +(T, G) is given

by supp−1(∆i

|V (T)|), where ∆i |V (T)| is the i-skeleton of ∆|V (T)|.

• The first tableau looks like this:

m − 3 m − 1 m − 2 p q n − 2 2n − 4 n − 3 H∗(Hom (Cm, Kn)) D0 D1 D2

d1 d1 d1 d1 d1 d1 d1 d1 d1 d1 d1 d2 d1

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Cohomology of complexes of cycles III. Definition. Let m, n, g ∈ N, Φm,n,g is the cubical complex whose cells are indexed by all possible collections of m sets, where each set either contains a single element of the circular n-set, or consists of a pair of two elements at distance 1; the sets are requested to be at minimal distance g, where the distance between two sets is defined as the minimum of the distances between their elements.

• Example. Cell indexing in Φ2,8,2:

5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2 5 1 7 4 3 6 8 2

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Cohomology of complexes of cycles IV.

• Definition. Let m, n, and g be natural numbers. The cubical

complex TFm,n,g is defined as follows: vertices of TFm,n,g are indexed by all possible (m, n)-torus fronts, whose horizontal legs have length at least g, here the length of the leg is taken to be the number of vertices it contains; the higher-dimensional cubes of TFm,n,g are indexed by all possible flips of (m, n)-torus fronts, whose members are vertices of TFm,n,g. The rows in the first tableau can be reinterpreted as computing homology of Φm,n,g = TFm,n−m,g.

(0, 0) millstones of T T millstones of the flip (5, 3)

One can grind torus front complexes until only thin fronts are left.

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Cohomology of complexes of cycles V.

• Theorem. (K., 2005)

For any integers m, n, such that m ≥ 5, n ≥ 4, we have

• H∗(Hom (Cm, Kn); Z) =

 

⌊(m−2)/3⌋

• t=1

At,m,n   ⊕ Bm,n, where At,m,n = Z(tn − 3t) ⊕ Z(tn − 3t + 1), if n is odd or m + t is odd, Z2(tn − 3t + 1), if n is even and m + t is even, and Bm,n =    Z2n−3(nk − m), if m = 3k, Z(nk − m + 2), if m = 3k + 1, Z(nk − m), if m = 3k − 1. Examples:

H∗(Hom (C6, K4); Z) = A1,6,4⊕B6,4 = Z(1)⊕Z(2)⊕Z13(2) = Z(1) ⊕ Z14(2);

H∗(Hom (C8, K6); Z) = A1,8,6 ⊕ A2,8,6 ⊕ B8,6 = Z(3) ⊕ Z(4) ⊕ Z2(7) ⊕ Z(10).

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Summary and Outlook

The fact that the Lov´ asz Conjecture is true implies that Hom (−, −) construction is interesting and produces nontrivial bounds for the chromatic number. It is known that the following more general conjecture is false: χ(G) ≥ χ(T) + conn Hom (T, G) + 1. Is truth somewhere in the middle? How much information about the graph colorings is contained in the algebro-topological invariants of the Hom -complexes?

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References

[1] D.N. Kozlov, Chromatic numbers, morphism complexes, and Stiefel- Whitney characteristic classes, in Geometric Combinatorics, IAS/ Park City Mathematics Series 14, American Mathematical Society, Providence, RI; Institute for Advanced Study, Princeton, NJ. [2] E. Babson, D.N. Kozlov, Proof of the Lov´ asz Conjecture, Annals of Mathematics, to appear. arXiv:math.CO/0402395 [3] E. Babson, D.N. Kozlov, Topological obstructions to graph colorings,

• Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 61–68.

[4] E. Babson, D.N. Kozlov, Complexes of graph homomorphisms, Israel J. Math., to appear. arXiv:math.CO/0310056 [5] D.N. Kozlov, Complexes of directed trees, J. Combin. Theory Ser. A 88 (1999), no. 1, 112–122. [6] D.N. Kozlov, A simple proof for folds on both sides in complexes of graph homomorphisms, Proc. Amer. Math. Soc., to appear. arXiv:math.CO/0408262 [7] S. ˇ Cuki´ c, D.N. Kozlov, Higher connectivity of graph coloring com- plexes, IMRN, to appear. arXiv:math.CO/0410335 [8] D.N. Kozlov, Simple homotopy type of some combinatorially defined complexes. arXiv:math.AT/0503613 [9] D.N. Kozlov, Cohomology of colorings of cycles. arXiv:math.AT/0507117