Covering map theory for graphs - Osaka City University€¦ · (Lov´asz [10] ) G ¬ å Ñ q ` |n...

Covering map theory for graphs

( )∗

1.V V × V E (x, y) ∈ V × V (x, y) ∈

E ⇒ (y, x) ∈ E (V,E)

G = (V,E) V V (G) G

E E(G) G,H G H

f : V (G) → V (H) (f × f)(E(G)) ⊂ E(H)

1.1. G,H G H

1.1

[13] [14] [15]

2.

[9]

V V ∆

σ ∈ ∆ τ ∈ 2V τ ⊂ σ τ ∈ ∆ v ∈ V{v} ∈ ∆ (V,∆) V (V,∆)∆ V ∆

∆ V (∆)

∆1 ∆2 ∆1 ∆2f : V (∆1) → V (∆2) σ ∈ ∆1 f(σ) ∈ ∆2

V R(V ) V R- R(V )R- R(V )

∆ ∆ σ ∆σ {∑

v∈σ avv ∈ R(V ) | av ≥ 0, v ∈ σ,∑

v∈σ av = 1}R(V (∆)) ∆

⋃σ∈∆ ∆σ R(V (∆))

∆ |∆| f : ∆1 → ∆2f R(V (∆1)) → R(V (∆2)) |f | : |∆1| → |∆1|

P c P

x, y ∈ c x ≤ y x ≥ y

( :254699)∗ 153-8914 3-8-1e-mail: [email protected]

第６０回トポロジーシンポジウム講演集　2013年8月　於大阪市立大学

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P P P

∆(P ) P |∆(P )| |P |∆ ∆ \ {∅} ∆

F (∆) ∆ |F (∆)| |∆|

f : ∆1 → ∆2f |f | : |∆1| → |∆2|

3. Lovász Kneser1.1

n n Kn V (Kn) = {1, · · · , n} E(Kn) = {(x, y) ∈V (Kn)2 | x ̸= y} G inf{n ≥ 0 | G Kn

} G G χ(G) χ(G)(graph coloring problem)

G

Lovász Lovász [10]

Kneser

3.1. G G v N(v) = {w ∈ V (G) | (v, w) ∈ E(G)}G N (G) {v ∈ V (G) | N(v) ̸= ∅}

N (G) = {σ ⊂ V (G) | ♯σ< +∞ G v σ ⊂ N(v) }

Lovász

3.2. (Lovász [10] ) G n (−1) N (G)n- 1 χ(G) ≥ n+ 3

Lovász N (G) Z2- L(G) (Lovasz) G Km L(G)

m Borsuk-Ulam 2

Kneser n, k n ≥ 2k V (Kn,k) = {σ ⊂{1, · · · , n} | ♯σ = k} E(Kn,k) = {(σ,τ ) | σ∩τ = ∅} Kneser

[8] Kneser χ(Kn,k) ≤ n−2k+2 χ(Kn,k) = n−2k+2

1 (−1)-2Sn (n+ 1)


12

(Kneser ) Lovász N (Kn,k) (n− 2k− 1)-3.2 Kneser Kneser Bárány [1]

Greene [7] Borsuk-Ulam

Matoušek [12]

Hom [2] [5] [9] [11] [17]

[18]

3.2 Hom

G H 3 Hom(G,H) C2r+1 (2r+1)-

5.1.(2)

3.3. (Babson-Kozlov [3]) G n (−1) rHom(C2r+1, G) n- χ(G) ≥ n+ 4

Hom Lovász Lovász

[17]

2000 Hom

4.[13] [14] [15] r

r- r- r-

r- r-

r-

r-

r- 5.5 r-

1.1

r Lovász r-

1- r-

(2r)- 7.1 r

r- Z2-

5. r- r-r r- r-

r- r-

G v ∈ V (G) i V (G) Ni(v)N1(v) = N(v) Ni+1(v) =

⋃w∈Ni(v) N(w)

p : G → H r- v ∈ V (G) 1 ≤ i ≤ r ip : Ni(v) → Ni(p(v))

3 [9] [6][13] [14]


13

5.1. r-

(1) G H G×H V (G×H) = V (G)×V (H) E(G×H) = {((x, y), (x′, y′)) | (x, x′) ∈ E(G), (y, y′) ∈ E(H) G

K2 ×G → G r r- Gχ(G) ≥ 3 K2 ×G

(2) n n- Cn V (Cn) = Z/nZ E(Cn) = {(x, x±1) | x ∈ Z/nZ} n ≥ 3 Cn n

K3 ∼= C3n 3 k p : Cnk → Cn (x mod.nk) &→ x mod.n

(i) n k = 1, 2 r

p r- n K2 ×Cn ∼= C2nk ≥ 3 k = 0 p (n− 1)- n-

(ii) n k = 1 r p

r- p (n/2) − 1-(n/2)-

(G, v) r- πr1(G, v)

5.2. n Ln V (Ln) = {0, 1, · · · , n} E(Ln) = {(x, y) | |x−y| = 1} (G, v) Ln G ϕϕ(0) = ϕ(n) = v (G, v) n (G, v)

L(G, v) ϕ,ψ ∈ L(G, v) (1),(2)r

(1) ϕ n ψ (n+2) x ∈ {0, 1, · · · , n}ϕ(i) = ψ(i) (i ≤ x) ϕ(i) = ψ(i+ 2)

(2)r ϕ ψ n x ∈ {0, 1, · · · , n}i ̸∈{ x, x+ 1, · · · , x+ r − 2} ϕ(i) = ψ(i)

(1) (2)r L(G, v) ≃r L(G, v)/ ≃r πr1(G, v)(G, v) r-

5.3. r-

(1) r = 1 (2)r ϕ = ψ G4 π11(G, v) G 1 CW

4 v ∈ V (G) (v, v) ∈ E(G)


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(2) ϕ l(ϕ) ϕ πr1(G, v) → Z/2Z,[ϕ] "→ (l(ϕ) mod.2) well-defined πr1(G, v)ev

πr1(G, v) α ∈ πr1(G, v) α ∈ πr1(G, v)ev α

r-

(3) r ≥ s ϕ ≃s ψ ⇒ ϕ ≃r ψ πs1(G, v) →πr1(G, v)

(4) α ∈ πr1(G, v) l(α) = inf{l(ϕ) | ϕ ∈ α} α,β ∈ πr1(G, v)dr(α,β ) = l(α−1β) dr πr1(G, v)

f : (G, v) → (H,w) dr(f∗α, f∗β) ≤ dr(α,β )dr(f∗α, f∗β) = dr(α,β ) (mod.2) α,β ∈ πr1(G, v)

5.4. r-

(1) n 3 πr1(Cn) r < n Z r ≥ nπr1(Cn) ∼= Z/2Z n 4 r < (n/2) πr1(Cn) ∼= Z

r ≥ (n/2) πr1(Cn)

(2) (1) π21(K3) ∼= Z 4 n π21(Kn) ∼= Z/2Z

r- r-

5.5. (G, v)

Xr : (G, v) r- (G, v)

Yr : πr1(G, v) πr1(G, v)

Xr → Yr, (p : (H,w) → (G, v)) "→ Im(πr1(p))

(1) r- p : (H,w) → (G, v) πr1(p) : πr1(H,w) → πr1(G, v)

(2) pi : (Hi, wi) → (G, v) (i = 1, 2) r- Hif : (H1, w1) → (H2, w2) p2 ◦ f = p1

Im(πr1(p1)) ⊂ Im(πr1(p2)) f

(3) πr1(G, v) Γ r- pΓ : (GΓ, vΓ) → (G, v) GΓIm(πr1(pΓ)) =Γ

5.6.

(1) (G, v) G χ(G) ≥ 3 K2 ×G(K2×G, (1, v)) → (G, v) r r-

r- πr1(G, v) πr1(G, v)ev


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(2) n 3 r < n πr1(Cn) ∼= ZkZ ⊂ Z r- Cnk → Cn, (x mod.nk) $→ (x mod.n)

(3) Kn n ≥ 4 π21(Kn) ∼= Z/2Z n ≥ 4 Kn2- Kn K2 ×Kn

6. r-r-

6.1. n 3 r n G G

Cn α ∈ πr1(G)l(αk) ≥ kn (l(α) 4.3(2) ) kπr1(G) χ(G) ≥ 3 πr1(G)

Proof. α ∈ πr1(G) f∗α ∈ πr1(Cn)l(f∗α) ≥ n πr1(Cn) ∼= Z l(f∗(α)k) = l(f∗(α))k

l(αn) ≥ l(f∗(α)k) ≥ nk

n = 3 C3 ∼= K36.2. G χ(G) = 3 π21(G)

6.3. 6.1

(1) G g0(G) = inf{2n+1 | n ≥ 0 C2n+1 G} g0(G) G g0(G) = sup{2n+1 | n ≥ 0

G C2n+1 } n,mn > m Cn Cm Cm Cn

g0(G) ≥ g0(G) g0(G)g0(G)

6.1 n,m n ≥ mGn,m g0(G) = n g0(Gn,m)

G g0(G) = 7 g0(G) = 5

(2) n Gn V (Gn) = {(i.j) ∈ Z2 | 0 ≤ i, j ≤ n}E(Gn) = {((i, j)(i′, j′)) | |i− i′| + |j − j′| = 1} Xnj ∈ {0, 1, · · · , n} (j, 0) (n, j) (n− j, n) (0, n− j) Gn

n χ(Xn) = 2 n Cn Xnχ(Xn) ≥ 3 π21(X3) ∼= Z/2Z n ≥ 5

n π21(Xn) ∼= Z/4Z n χ(Xn) ≥ 4


16

! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !

1

4 χ(Xn) = 4

n 5 G χ(G) > 2 π21(G) ∼= Z/2ZG Km (m ≥ 4) m ≥ 2k+ 2

m, k Kneser Km,k Kneser SKm,k7.6 G Xn

2- G Xnf : G → Xn α ∈ π21(G) ∼= Z/2Zf∗α π21(Xn) ≃ Z/4Z f∗(α)2 = 1

π21(Xn)

(3) Kneser K2k+1,k 3 K2k+1,k C3(∼= K3)K2k+1,k C5

g0(K2k+1,k) = 2k+1 K2k+1,k 2k+1 π31(Kn,k)

2 5

(4) G Cnr < n πr1(G)

l(αk) ≥ nk

n 5 Xn (2) π21(Xn) ∼=Z/4Z Xn 2- X̃n X̃n Z/4Z

Z/4Z τ : X̃n → X̃nỸn V (Ỹn) = V (X̃n)× Z E(Ỹn) = {((x, i), (y, j)) | (i = j (x, y) ∈ E(G))

(x = y |i− j| = 1) } Ỹn ZZ (x, i) (→ (τ(x), i + 2)

Ỹn π21(Yn) ∼= Zα l(α) = n+ 2 l(α4) = 8 < 3 · 4 Yn K3

5π31(K2k+1,k) ∼= Z/2Z


17

χ(Yn) > 3 Yn 4

χ(Yn) = 4

7. r-[15] r- Lovász

r- (2r)-

7.1. G r- (r-neighborhood complex)

Nr(G) = {σ ⊂ V (G) | ♯σ< ∞ v ∈ V (G) σ ⊂ Nr(v) }

1- Lovász N (G)

7.2. r (G, v) N(v) ̸= ∅ π2r1 (G, v)ev ∼=π1(Nr(G), v)

7.3. G χ(G) > 2 n 3 G

Cn H1(N(n−1)/2(G);Z) Z

G Cn f πn−11 (G) →

πn−11 (Cn) ∼= Z χ(G) > 2 πn−11 (G)πn−11 (f) π

n−11 (G)ev π

n−11 (G) 2 π

n−11 (G)ev

Z πn−11 (G)ev Z6.2

7.4. p : G → H (2r)- Nr(p) : Nr(G) → Nr(H)

6.2 r- (2r)-

7.5. r (G, v) G N(v) ̸= ∅

(1) χ(G) = 2 (Nr(G), v)(G, v)

(2r)-

(2) χ(G) ≥ 3 (Nr(G), v)(K2 × G, (1, v))

(2r)-

7.6. 6.2

(1) G G+

V (G+) = V (G) *{∗} E(G+) = E(G) ∪ (V (G)×{∗} ) ∪ ({∗}× V (G))


18

N (G+) N (G)[5] χ(G) ≥ 3 G N (G)

N (G) π21(G) ∼= Z/2ZG+ 2- G+ K2 ×G+

(2) Lovász Kn,k n ≥ 2k N (Kn,k) (n− 2k − 1)-Kneser SKn,k

V (SKn,k) = {σ ⊂ {1, · · · , n} | ♯σ = k i ∈ σ i+ 1 ̸∈ σ }

E(SKn,k) = {(σ,τ ) | σ ∩ τ = ∅}

Schriver N (SKn,k) (n − 2k − 1)- 6

χ(SKn,k) = n − 2k + 2 n ≥ 2k + 2 Kn,kSKn,k 2- Z/2Z 2-

K2 ×Kn,k K2 × SKn,k

r Lovász r-

7.7. r G Br(G)

{(σ,τ ) | σ,τ V (G) v ∈ σ τ ⊂ Nr(v)}

(σ,τ ) ≤ (σ′, τ ′) ⇔ σ ⊂ σ′ τ ⊂ τ ′

∆ F∆ 2

7.8. G Br(G) → FN (G), (σ,τ ) -→ σBr(G) Nr(G)

r g0(G) > r Br(G) Z2- G Hg0(G), g0(H) > r Br(G) Br(H) Stiefel-Whitney

0 G H

7.9. n, k, r n > 2k (n − 2k)r = k − 1v ∈ V (Kn,k) N2r−1(v) = V (Kn,k) − {v}

g0(Kn,k) = 2r+ 1 N2r−1(Kn,k) (♯V (Kn,k)− 2) = ((nk

)− 2)

7.10. n, k, r n > 2k (n− 2k)r = k − 1(2r+1) G ♯V (G) <

(nk

)Kn,k

G

r-

6 Björner Longueville [4] N (SKn,k) Sn−2k


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[1] I. Barany, A short proof of Kneser’s conjecture, J. Combin. Ser. A 25 (3) 325-326, (1978)

[2] E. Babson, D. Kozlov, Complexes of graph homomorphisms, Israel J. Math. 152, 285-312(2006)

[3] E. Babson, D. Kozlov, Proof of the Lovász conjecture, Ann. Math. 165 (3) 965-1007(2007)

[4] A. Björner, M. Longueville, Neighborhood complexes of stable Kneser graphs, Combina-torica, 23 (1):23-34, 2003.

[5] P. Csorba, Homotopy types of box complexes. Combinatorica 27 (2007) (6):669-682.

[6] A. Dochtermann, Hom complexes and homotopy theory in the category of graphs, Euro-pean J. Combin. 30 (2) 490-509 (2009)

[7] J. Greene, A new short proof of Kneser’s conjecture, Amer. Math. Monthly 109 (2002),no. 10, pp. 918-920.

[8] M. Kneser, Aufgabe 360, Jber. Deutsch. Math.-Verein. 58, (1955/56), 2 Abt., 27

[9] D.N. Kozlov Combinatorial algebraic topology, Algorithms and Computation in Mathe-matics 21, Springer, Berlin, 2008

[10] L. Lovász, Kneser’s conjecture, chromatic number and homotopy, J. Comb. Theory, Ser.A, 25 (1978), 319-324

[11] J. Matoušek, G. Ziegler, Topological lower bounds for the chromatic number, Jahres-bericht der DMV 106, 71-90, 2004

[12] J. Matoušek, A combinatorial proof of Kneser’s conjecture, Combinatorica 24 (2004),no. 1, pp. 163-170

[13] T. Matsushita, Fundamental groups of neighborhood complexes, arXiv:1210.2803

[14] T. Matsushita, Generalized covering map theory of graphs, arXiv:1301.7217

[15] T. Matsushita, Generalization of neighborhood complexes, arXiv:1305.2503

[16] A. Schriver, Vertex-critical subgraphs of Kneser graphs, Nieuw Arch. Wisk. III, Ser.,26:454-461, 1978

[17] C. Schultz, Graph colourings, spaces of edges and spaces of circuits, Adv. Math. 221(6):1733-1756, 2009

[18] R. Živaljević, WI-posets, graph complexes and Z2-equivalences, J. Combin. Theory Ser.A 111 (2005), no. 2, pp. 204-223


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Covering map theory for graphs - Osaka City University€¦ · (Lov´asz [10] ) G ¬ å Ñ q ` |n...

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