A universality argument for graph homomorphisms cka 2 and Yangjing - - PowerPoint PPT Presentation

A universality argument for graph homomorphisms

Jiˇ r´ ı Fiala 1, Jan Hubiˇ cka 2 and Yangjing Long 3

1 Charles University, Czech Republic 2 University of Calgary, Canada 3 Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Terminology

Partial order . . . a reflexive, antisymmetric and transitive relation ≤ over a countable set P Past-finite . . . every down set ↓ x = {y : y ≤ x} is finite. Future-finite . . . every up set ↑ x = {y : y ≥ x} is finite. Examples of past-finite orders:

◮ N ordered by divisibility ◮ Finite subsets Pfin(A) of a countable set A ordered by inclusion

Two future-finite orders:

◮ (ConnGraph, ≤B) and (ConnGraph, ≤S)

Observe: (P, ≤) is past-finite ⇐ ⇒ (P, ≥) is future-finite. Observe: ↓ x ⊆ ↓ y ⇐ ⇒ x ≤ y. Corollary: A past-finite (P, ≤) is a suborder of (Pfin(P), ⊆) (via the embedding x → ↓ x).

Universality

Definition: A partial order is        finite-universal past-finite-universal future-finite-universal universal        if it contains any        finite past-finite future-finite countable       

• rder as a suborder.

Proposition: For any countably infinite A:

◮ (Pfin(A), ⊆) is past-finite-universal. ◮ (Pfin(A), ⊇) is future-finite-universal.

. . . w.l.o.g. consider only (P, ≤) where P ⊆ A, then use x → ↓ x. Example: (Pfin(P), ⊇) is future-finite-universal, where P are all odd primes.

Definition: The subset order (Pfin(Q), ≤dom

Q

) of (Q, ≤Q) is given by X ≤dom

Q

Y iff ∀x ∈ X ∃y ∈ Y : x ≤Q y. Theorem ”[Hedrl´ ın 1969]”: If (F, ≤F) is future-fin.-universal, then (Pfin(F), ≤dom

F

) is universal. Proof: Given any countable (P, ≤P), w.l.o.g. P ⊆ N. Then:

• 1. decompose ≤P into
• x ≤f y

iff x ≤P y and x ≤ y x ≤b y iff x ≤P y and x ≥ y. . . . (P, ≤f ) is past-finite and (P, ≤b) is future-finite.

• 2. find an embedding e : (P, ≤b) → (F, ≤F).
• 3. argue that g(x) = {e(y) : y ≤f x}

is an embedding of (P, ≤P) in (Pfin(F), ≤dom

F

).

Example with (Pfin(P), ⊇) as (F, ≤F)

(P, ≤P)

The given order (P, ≤P),

Example with (Pfin(P), ⊇) as (F, ≤F)

(P, ≤P) 3 7 5 11

label P by P ⊂ N

Example with (Pfin(P), ⊇) as (F, ≤F)

(P, ≤P) (P, ≤f ) 3 7 5 11 3 7 5 11 3 7 5 11 (P, ≤b)

decompose ≤P into

• x ≤f y

iff x ≤P y and x ≤ y x ≤b y iff x ≤P y and x ≥ y.

Example with (Pfin(P), ⊇) as (F, ≤F)

(P, ≤P) (P, ≤f ) 3 7 5 11 3 7 5 11 3 7 5 11 (P, ≤b) embedding in (Pfin(P), ⊇) {3} {3, 5, 7} {5} {5, 11} e

find an embedding e : (P, ≤b) → (F, ≤F)

Example with (Pfin(P), ⊇) as (F, ≤F)

(P, ≤P) (P, ≤f ) 3 7 5 11 3 7 5 11 3 7 5 11 (P, ≤b) embedding in (Pfin(P), ⊇) {3} {3, 5, 7} {5} {5, 11} e {{3}} {{3, 5, 7}} {{3}, {5}} {{3, 5, 7}, {5, 11}} embedding in (Pfin(Pfin(P)), ⊇dom

Pfin(P))

g

define embedding by g(x) = {e(y) : y ≤f x}

Example with (Pfin(P), ⊇) as (F, ≤F)

(P, ≤P) (P, ≤f ) 3 7 5 11 3 7 5 11 3 7 5 11 (P, ≤b) embedding in (Pfin(P), ⊇) {3} {3, 5, 7} {5} {5, 11} e {{3}} {{3, 5, 7}} {{3}, {5}} {{3, 5, 7}, {5, 11}} embedding in (Pfin(Pfin(P)), ⊇dom

Pfin(P))

g

Recall: X ⊇dom

Pfin(P) Y

iff ∀X ∈ X ∃Y ∈ Y s.t. X ⊇ Y Hence {{3}} is incomparable with {{3, 5, 7}, {5, 11}}.

Indeed (Pfin(Pfin(P)), ⊇dom

Pfin(P)) ⊂ (Pfin(N), ←

− | dom

N

)

Let a = X, b = Y , A = { X, X ∈ X}, B = { Y , Y ∈ Y} then X ⊇dom

Pfin(P) Y

⇐ ⇒ ∀X ∈ X ∃Y ∈ Y : X ⊇ Y ⇐ ⇒ ∀a ∈ A ∃b ∈ B : a is divided by b ⇐ ⇒ A← − | dom

N

B

(P, ≤P) {{3}} {{3, 5, 7}} {{3}, {5}} {{3, 5, 7}, {5, 11}} embedding in (Pfin(Pfin(P)), ⊇dom

Pfin(P))

{3} {105} {3, 5} {105, 55} embedding in (Pfin(N), ← − |

dom N

)

Consequences on homomorphism orders

Theorem: Collections of directed cycles ordered by homomorphisms are universal.

(P, ≤P) {3} {105} {3, 5} {105, 55} embedding in (Pfin(N), ← − |

dom N

)

• C105
• C105 ∪

C55 embedding in (DiCycles, ≤Hom)

Consequences on homomorphism orders

Theorem: Collections of directed cycles ordered by homomorphisms are universal. Corrollary: Homomorphism order is universal on graphs that are

◮ maximum degree 3, ◮ planar, ◮ have treewidth at most 4, etc.

replace all by in each Ck

Many other directions

Choose mappings M monomorphisms F full homomorphisms E embeddings VS vertex surjective homomorphisms ES edge surjective homomorphisms S surjective homomorphisms LB locally bijective homomorphisms LI locally injective homomorphisms LS locally surjective homomorphisms . . . . . . . . . Goal: Classify

◮ Universality

— if possible on a narrow subclass

◮ Cores ◮ Density ◮ Gaps

. . . . . . . . . Directed graphs could be also considered.

Coverings and their variants

Definition: A homomorphism f : G → H is a graph covering if f acts bijectively between N(u) and N(f (u)) for all u ∈ VG.

B

If f : G → H acts always injectively between N(u) and N(f (u)) then is a partial covering. If f acts locally surjectively then it is a role assignment.

I S

Degree refinement / equitable partition

Definition: A partition of VG into B1, . . . , Bk is called a degree partition if u, v ∈ Bi then |N(u) ∩ Bj| = |N(v) ∩ Bj| for all j. (Also known as an equitable partition.) The unique partition with minimum number of classes can be computed iteratively and is called degree refinement.

(3) (2) (1) (1) (1) (0,1,2) (1,0,1) (1,0,0) (1,0,0) (0,1,0) (0,1,2,0) (1,0,0,1) (1,0,0,0) (0,1,0,0) drm(G) =     1 2 1 1 1 1    

There is a canonical ordering of the classes B1, . . . , Bk, hence the constants mi,j = |N(u) ∩ Bj| for u ∈ Bi can be arranged into a unique degree refinement matrix drm(G). Folklore: On connected graphs: if G

B

− → H then drm(G) = drm(H).

Homomorphisms as orders

View G → H as G ≤ H, it is a transitive and reflexive relation. Similarly define ≤B, ≤I and ≤S Theorem [Fiala, Maxov´ a, 2006]: (ConnGraph, ≤B) = (ConnGraph, ≤I) ∩ (ConnGraph, ≤S) Observe: (ConnGraph, ≤B) is a disjoint union of orders. Graphs in the same part have the same degree refinement matrix. What about order properties like universatity, density, cores, gaps, dualities, . . . of these orders? Observe: (ConnGraph, ≤B) and (ConnGraph, ≤S) are not dense since G ≤B H or G ≤S H imply |VG| ≥ |VH|.

Consequences on covering orders

Theorem: Collections of cycles ordered by coverings are universal.

(P, ≤P) {3} {105} {3, 5} {105, 55} embedding in (Pfin(N), ← − |

dom N

) C105 C105 ∪ C55 embedding in (Cycles, ≤B)

Corollary: (Cycles, ≤I) and (Cycles, ≤S) are universal orders. Observe: drm could be also prescribed, except for forests

Consequences on covering orders

Theorem: Collections of cycles ordered by coverings are universal.

(P, ≤P) {3} {105} {3, 5} {105, 55} embedding in (Pfin(N), ← − |

dom N

) C105 C105 ∪ C55 embedding in (Cycles, ≤B)

Corollary: (Cycles, ≤I) and (Cycles, ≤S) are universal orders. Observe: drm could be also prescribed, except for forests Theorem: (ConnGraph, ≤I) is a universal order.

Density

Theorem: On connected graphs of minimum degree 2, distinct from cycles: If G <I H and drm(G) = drm(H), then there exists F, such that G <I F <I H and drm(G) = drm(F) = drm(H).

u f (u) G H

Density

Theorem: On connected graphs of minimum degree 2, distinct from cycles: If G <I H and drm(G) = drm(H), then there exists F, such that G <I F <I H and drm(G) = drm(F) = drm(H).

u f (u) G H u F

Density

Theorem: On connected graphs of minimum degree 2, distinct from cycles: If G <I H and drm(G) = drm(H), then there exists F, such that G <I F <I H and drm(G) = drm(F) = drm(H).

u f (u) G H u F

Density

Theorem: On connected graphs of minimum degree 2, distinct from cycles: If G <I H and drm(G) = drm(H), then there exists F, such that G <I F <I H and drm(G) = drm(F) = drm(H).

u f (u) G H u F

Density

Theorem: On connected graphs of minimum degree 2, distinct from cycles: If G <I H and drm(G) = drm(H), then there exists F, such that G <I F <I H and drm(G) = drm(F) = drm(H).

u f (u) G H u F