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

A universality argument for graph homomorphisms cka 2 and Yangjing Long 3 ı Fiala 1 , Jan Hubiˇ Jiˇ r´ 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 P fin ( 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 ( P fin ( P ) , ⊆ ) (via the embedding x → ↓ x ).

Universality Definition: A partial order is    finite  finite-universal         past-finite-universal past-finite     if it contains any future-finite-universal future-finite         universal countable     order as a suborder. Proposition: For any countably infinite A : ◮ ( P fin ( A ) , ⊆ ) is past-finite-universal. ◮ ( P fin ( A ) , ⊇ ) is future-finite-universal. . . . w.l.o.g. consider only ( P , ≤ ) where P ⊆ A , then use x → ↓ x . Example: ( P fin ( P ) , ⊇ ) is future-finite-universal, where P are all odd primes.

Definition: The subset order ( P fin ( Q ) , ≤ dom ) of ( Q , ≤ Q ) is given Q by X ≤ dom Y iff ∀ x ∈ X ∃ y ∈ Y : x ≤ Q y . Q Theorem ”[Hedrl´ ın 1969]”: If ( F , ≤ F ) is future-fin.-universal, then ( P fin ( F ) , ≤ dom ) is universal. F Proof: Given any countable ( P , ≤ P ), w.l.o.g. P ⊆ N . Then: � x ≤ f y iff x ≤ P y and x ≤ y 1. decompose ≤ P into 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 ( P fin ( F ) , ≤ dom ). F

Example with ( P fin ( P ) , ⊇ ) as ( F , ≤ F ) ( P , ≤ P ) The given order ( P , ≤ P ),

Example with ( P fin ( P ) , ⊇ ) as ( F , ≤ F ) ( P , ≤ P ) 5 3 11 7 label P by P ⊂ N

Example with ( P fin ( P ) , ⊇ ) as ( F , ≤ F ) ( P , ≤ b ) 5 3 11 ( P , ≤ P ) 5 7 3 11 ( P , ≤ f ) 7 5 3 11 7 � x ≤ f y iff x ≤ P y and x ≤ y decompose ≤ P into x ≤ b y iff x ≤ P y and x ≥ y .

Example with ( P fin ( P ) , ⊇ ) as ( F , ≤ F ) ( P , ≤ b ) embedding in ( P fin ( P ), ⊇ ) 5 { 5 } e 3 11 { 3 } { 5, 11 } ( P , ≤ P ) 5 7 { 3, 5, 7 } 3 11 ( P , ≤ f ) 7 5 3 11 7 find an embedding e : ( P , ≤ b ) → ( F , ≤ F )

Example with ( P fin ( P ) , ⊇ ) as ( F , ≤ F ) ( P , ≤ b ) embedding in ( P fin ( P ), ⊇ ) 5 { 5 } e 3 11 { 3 } { 5, 11 } ( P , ≤ P ) 5 7 { 3, 5, 7 } g 3 11 embedding in ( P fin ( P fin ( P )), ⊇ dom ( P , ≤ f ) P fin ( P ) ) 7 5 {{ 3 } , { 5 }} 3 11 {{ 3 }} {{ 3, 5, 7 } , { 5, 11 }} 7 {{ 3, 5, 7 }} define embedding by g ( x ) = { e ( y ) : y ≤ f x }

Example with ( P fin ( P ) , ⊇ ) as ( F , ≤ F ) ( P , ≤ b ) embedding in ( P fin ( P ), ⊇ ) 5 { 5 } e 3 11 { 3 } { 5, 11 } ( P , ≤ P ) 5 7 { 3, 5, 7 } g 3 11 embedding in ( P fin ( P fin ( P )), ⊇ dom ( P , ≤ f ) P fin ( P ) ) 7 5 {{ 3 } , { 5 }} 3 11 {{ 3 }} {{ 3, 5, 7 } , { 5, 11 }} 7 {{ 3, 5, 7 }} Recall: X ⊇ dom P fin ( P ) Y iff ∀ X ∈ X ∃ Y ∈ Y s.t. X ⊇ Y Hence {{ 3 }} is incomparable with {{ 3 , 5 , 7 } , { 5 , 11 }} .

P fin ( P ) ) ⊂ ( P fin ( N ) , ← − Indeed ( P fin ( P fin ( P )) , ⊇ dom | dom ) N Let a = � X , b = � Y , A = { � X , X ∈ X} , B = { � Y , Y ∈ Y} then X ⊇ dom P fin ( P ) Y ⇐ ⇒ ∀ X ∈ X ∃ Y ∈ Y : X ⊇ Y ⇐ ⇒ A ← − | dom ∀ a ∈ A ∃ b ∈ B : a is divided by b ⇐ ⇒ B N embedding in ( P fin ( N ), ← − dom embedding in ( P fin ( P fin ( P )), ⊇ dom ( P , ≤ P ) P fin ( P ) ) | ) N {{ 3 } , { 5 }} { 3, 5 } {{ 3 }} {{ 3, 5, 7 } , { 5, 11 }} { 3 } { 105, 55 } {{ 3, 5, 7 }} { 105 }

Consequences on homomorphism orders Theorem: Collections of directed cycles ordered by homomorphisms are universal. embedding in ( P fin ( N ), ← − dom ( P , ≤ P ) | ) embedding in ( DiCycles , ≤ Hom ) N { 3, 5 } � C 105 ∪ � { 3 } { 105, 55 } C 55 � { 105 } C 105

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 � C k

Many other directions Choose mappings Goal: Classify M monomorphisms ◮ Universality F full homomorphisms — if possible on E embeddings a narrow subclass VS vertex surjective homomorphisms ◮ Cores ES edge surjective homomorphisms ◮ Density S surjective homomorphisms ◮ Gaps LB locally bijective homomorphisms . . . LI locally injective homomorphisms . . . LS locally surjective homomorphisms . . . . . . . . . . . . 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 ∈ V G . 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 V G into B 1 , . . . , B k is called a degree partition if u , v ∈ B i then | N ( u ) ∩ B j | = | N ( v ) ∩ B j | 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) (0,1,2) (1,0,1) (0,1,2,0) (1,0,0,1) drm( G ) =   0 1 2 0 1 0 0 1     1 0 0 0   0 1 0 0 (1) (1) (1) (1,0,0) (1,0,0) (0,1,0) (1,0,0,0) (0,1,0,0) There is a canonical ordering of the classes B 1 , . . . , B k , hence the constants m i , j = | N ( u ) ∩ B j | for u ∈ B i can be arranged into a unique degree refinement matrix drm( G ). B Folklore: On connected graphs: if G − → 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 | V G | ≥ | V H | .

Consequences on covering orders Theorem: Collections of cycles ordered by coverings are universal. embedding in ( P fin ( N ), ← − dom ( P , ≤ P ) | ) embedding in ( Cycles , ≤ B ) N { 3, 5 } { 3 } { 105, 55 } C 105 ∪ C 55 { 105 } C 105 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. embedding in ( P fin ( N ), ← − dom ( P , ≤ P ) | ) embedding in ( Cycles , ≤ B ) N { 3, 5 } { 3 } { 105, 55 } C 105 ∪ C 55 { 105 } C 105 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 ). H G f ( u ) u

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 ). H G f ( u ) u F u

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 ). H G f ( u ) u F u

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 ). H G f ( u ) u F u

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 ). H G f ( u ) u F u