Universality of Intervals of Line Graph Partial Order cka 2 and - - PowerPoint PPT Presentation

Universality of Intervals of Line Graph Partial Order

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

1 Charles University, Czech Republic 2 University of Calgary, Canada 3 Shanghai Jiao Tong University

A new Universality argument of Homomorphism Orders

Outline

A new Universality argument of Homomorphism Orders What is universality and what are homomorphism orders? Universality of homomorphism order Our new agrument Application: Line Graph Intervals Why and what are line graph intervals? Are line graph intervals universal? What we do not know?

A new Universality argument of Homomorphism Orders

Universality

Definition

A partial order (P, ≤P) can be embedded into a partial order (Q, ≤Q) if there exists a mapping f from (P, ≤P) to (Q, ≤Q) such that a ≤P b if and only if f (a) ≤Q f (b).

Figure : No embeddings

Definition

A partial order P is universal, if any countable order can be embedded into P.

A new Universality argument of Homomorphism Orders

Why study universality?

If a partial order is universal, we can answer many questions like:

◮ whether there is an infinite chain. ◮ whether there is an infinite antichain (independent elements). ◮ given a particular partial order, whether there is an embedding

into it.

A new Universality argument of Homomorphism Orders

Homomorphism Orders (1980s):

◮ DiGraphs is the class of all finite directed graphs. ◮ Define (DiGraphs, ≤H): for G, H ∈ DiGraphs, G ≤H H iff

G → H.

◮ (DiGraphs, ≤H) is a quasi-order:

◮ The relation ≤H is reflexive (identity is a homomorphism). ◮ The relation ≤H transitive (composition of two

homomorphisms is still a homomorphism).

A new Universality argument of Homomorphism Orders

Homomorphism Orders (1980s):

◮ DiGraphs is the class of all finite directed graphs. ◮ Define (DiGraphs, ≤H): for G, H ∈ DiGraphs, G ≤H H iff

G → H.

◮ (DiGraphs, ≤H) is a quasi-order:

◮ The relation ≤H is reflexive (identity is a homomorphism). ◮ The relation ≤H transitive (composition of two

homomorphisms is still a homomorphism).

◮ Turn the quasi-order to a partial order: choose a particular

representative for each equivalence class, the cores fits perfectly the purpose.

◮ We denote the partial order the same way as the quasi-order,

by (DiGraphs, ≤H), where DiGraphs is restricted to cores.

A new Universality argument of Homomorphism Orders

Is the homomorphism order universal?

Theorem (Hedrl´ ın et al., 1980s)

The partial order (DiGraphs, ≤H) is universal. The proof is complicated.

◮ We consider universality of homomorphism order on graph

classes, like

• n simple graphs, cycles, paths, perfect graphs, planar graphs,

etc.

◮ Constrained homomorphism

Monomorphisms, surjective homomorphisms, full homomorphisms, locally injective homomorphisms.

A new Universality argument of Homomorphism Orders

DiPath: The class of oriented paths.

Theorem (Hubiˇ cka, Neˇ setˇ ril, 2003)

The homomorphism order on the class of oriented paths (DiPath, ≤H) is universal.

Figure : Zig Zag

A new Universality argument of Homomorphism Orders

DiPath: The class of oriented paths.

Theorem (Hubiˇ cka, Neˇ setˇ ril, 2003)

The homomorphism order on the class of oriented paths (DiPath, ≤H) is universal.

Figure : Zig Zag

A new Universality argument of Homomorphism Orders

Why Zig Zag is useful?

Theorem: Directed paths ordered by homomorphisms are universal. Advantage: Corrollary: Homomorphism order is universal on graphs that are

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

replace all by in each Pk

A new Universality argument of Homomorphism Orders

Why Zig Zag is useful?

Theorem: Directed paths ordered by homomorphisms are universal. Advantage: Corrollary: Homomorphism order is universal on graphs that are

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

replace all by in each Pk

Disadvantage: However, there are some problem which cannot build from zig zag easily, for example, locally injective homomorphism, if you because paths fliping is not locally injective. So we need new techniques.

A new Universality argument of Homomorphism Orders

We provide a new and significantly easier method to prove the universality. DiCycles: The class of graphs formed by finite disjoint union of clockwise oriented cycles. · · ·

− → C3 − → C4 − → C5 − → C6

Theorem (Fiala, Hubiˇ cka, L., 2012)

The partial order (DiCycles, ≤H) is universal.

◮ This proof is significantly easier than other proofs. ◮ It gives a new and simple universal class, this can be easily

applied to prove the universality of other orders.

◮ More applications: recently Neˇ

setˇ ril and Hubiˇ cka applied it to prove the fractal property.

A new Universality argument of Homomorphism Orders

Theorem (Fiala, Hubiˇ cka, L., 2012)

The partial order (DiCycles, ≤H) is universal.

◮ It is a well-known result that any finite order can be represented by

finite sets ordered by the containedness.

◮ We generalize it on up-finite order (infinite):

Any up-finite order can be represented by sets ordered by the containedness, represent every element by its up-set.

3 5 7 11

◮ Sets containedness contains no infinite increasing chain.

Idea: to use the sets of sets to represent a partial order.

A new Universality argument of Homomorphism Orders

On-line Embedding Game

◮ Bob has a secret partial order, each round he gives Alice one element and

the orders to the previous elements.

◮ If Alice recover Bob’s partial order on sets of sets, she wins.

Alice’s wining strategy:

◮ Each round gives a set and a set of sets, when given order is in

forwarding, add it to the set of sets, when given order is in backwarding, add it to set.

◮ Order the sets of sets by set dominate: A ≤ B iff for any a ∈ A there is a

set b ∈ B such that a ⊇ b.

3 5 7 11

3 {3} and {{3}}

A new Universality argument of Homomorphism Orders

On-line Embedding Game

◮ Bob has a secret partial order, each round he gives Alice one element and

the orders to the previous elements.

◮ If Alice recover Bob’s partial order on sets of sets, she wins.

Alice’s wining strategy:

◮ Each round gives a set and a set of sets, when given order is in

forwarding, add it to the set of sets, when given order is in backwarding, add it to set.

◮ Order the sets of sets by set dominate: A ≤ B iff for any a ∈ A there is a

set b ∈ B such that a ⊇ b.

3 5 7 11

3 {3} and {{3}} 5 and 3 → 5 {5} and {{3}, {5}}

A new Universality argument of Homomorphism Orders

On-line Embedding Game

◮ Bob has a secret partial order, each round he gives Alice one element and

the orders to the previous elements.

◮ If Alice recover Bob’s partial order on sets of sets, she wins.

Alice’s wining strategy:

◮ Each round gives a set and a set of sets, when given order is in

forwarding, add it to the set of sets, when given order is in backwarding, add it to set.

◮ Order the sets of sets by set dominate: A ≤ B iff for any a ∈ A there is a

set b ∈ B such that a ⊇ b.

3 5 7 11

3 {3} and {{3}} 5 and 3 → 5 {5} and {{3}, {5}} 7 and 7 → 3, 7 → 5 {3, 5, 7} and {{3, 5, 7}}

A new Universality argument of Homomorphism Orders

On-line Embedding Game

◮ Bob has a secret partial order, each round he gives Alice one element and

the orders to the previous elements.

◮ If Alice recover Bob’s partial order on sets of sets, she wins.

Alice’s wining strategy:

◮ Each round gives a set and a set of sets, when given order is in

forwarding, add it to the set of sets, when given order is in backwarding, add it to set.

◮ Order the sets of sets by set dominate: A ≤ B iff for any a ∈ A there is a

set b ∈ B such that a ⊇ b.

3 5 7 11

{{3}} {{3, 5, 7}} {{3}, {5}} {{3, 5, 7}, {5, 11}}

WIN! 3 {3} and {{3}} 5 and 3 → 5 {5} and {{3}, {5}} 7 and 7 → 3, 7 → 5 {3, 5, 7} and {{3, 5, 7}} 11 and 11 → 5, 7 → 11 {11, 5} and {{3, 5, 7}, {5, 11}}

A new Universality argument of Homomorphism Orders

We show the proof in two layers, first represent any partial order by another order on sets, then transfer it on DiCycles graphs.

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

dom N

)

• C105
• C105 ∪

C55 embedding in (DiCycles, ≤Hom)

Application: Line Graph Intervals

Outline

A new Universality argument of Homomorphism Orders What is universality and what are homomorphism orders? Universality of homomorphism order Our new agrument Application: Line Graph Intervals Why and what are line graph intervals? Are line graph intervals universal? What we do not know?

Application: Line Graph Intervals

Why and what are line graph intervals?

◮

Figure : A graph and its line graph

◮ If the maximal degree of G is n, then

• 1. Kn → L(G).
• 2. (Vising Theorem) The chromatic numer of L(G) is bounded by

n + 1, i.e., L(G) → Kn+1.

◮ From Vising theorem, if a graph G has maximal degree n, then

Kn → L(G) → Kn+1. A line graph L(G) is in the line graph interval [Kn, Kn+1]L if Kn → L(G) → Kn+1.

Question

[Roberson’s thesis 2013 Waterloo] Whether ([Kn, Kn+1]L, ≤H) are universal?

Application: Line Graph Intervals

Let’s play with it!

([K1, K2]L, ≤H) = {K1, K2} ([K2, K3]L, ≤H)= {All odd Cycles}. Because all the graphs have maximal degree 2, and are line graphs, cores.

Application: Line Graph Intervals

Let’s play with it!

([K1, K2]L, ≤H) = {K1, K2} ([K2, K3]L, ≤H)= {All odd Cycles}. Because all the graphs have maximal degree 2, and are line graphs, cores. ([K3, K4]L, ≤H)= core + contais a triangle + 4-colorable + line graphs +

• riginal graphs have maximal degree 3.

Aim: To construct a series of graphs L(Mk) ∈ [K3, K4]L such that the homomorphisms among these graphs L(Mk) have similar behavior as the homomorphisms among oriented cycles.

Application: Line Graph Intervals

Gadget: Dragon? Dragon!

a b u v w

Application: Line Graph Intervals

Gadget: Dragon? Dragon!

a b u v w

Dragon(Czech)= Kate(English) Dragon(English)= 龍(Chinese)

Application: Line Graph Intervals

Construction

Sunlet Sk is cycle Ck attaching n pendant edges. Construct {Mk} by replacing each edge of sunlet Sk by a dragon.

a b u v w

Dragon D3 and its line graph

Application: Line Graph Intervals

Construction

Sunlet Sk is cycle Ck attaching n pendant edges. Construct {Mk} by replacing each edge of sunlet Sk by a dragon.

a b u v w

Dragon D3 and its line graph Sunlet S5

Application: Line Graph Intervals

Construction

Sunlet Sk is cycle Ck attaching n pendant edges. Construct {Mk} by replacing each edge of sunlet Sk by a dragon.

c a b d e f

Dragon D3 and its line graph M5 L(M5)

Application: Line Graph Intervals

Why it works

L(G 3

5 )

Kd+1 : e L(Kd+1 : e) is a core impossible I. II. III.

Application: Line Graph Intervals

Towards line graphs

Theorem: [Fractal Property on Line Graphs, Fiala, Hubiˇ

cka, L., 2014] ([Kn, Kn+1]L, ≤H) is universal for any n ≥ 3. For every integer d ≥ 3 the line graphs of graphs with ∆(G) = d are universal when ordered by homomorphisms.

Application: Line Graph Intervals

Towards line graphs

Theorem: [Fractal Property on Line Graphs, Fiala, Hubiˇ

cka, L., 2014] ([Kn, Kn+1]L, ≤H) is universal for any n ≥ 3. For every integer d ≥ 3 the line graphs of graphs with ∆(G) = d are universal when ordered by homomorphisms.

c a b d e f

Figure : Dragons and 龍(dragon) in Oracle

What we do not know?

Outline

A new Universality argument of Homomorphism Orders What is universality and what are homomorphism orders? Universality of homomorphism order Our new agrument Application: Line Graph Intervals Why and what are line graph intervals? Are line graph intervals universal? What we do not know?

What we do not know?

What we do not know?

◮ All the orders we considered which are not universality due to

some simple reasons, like missing infinitely antichain, decreasing chain, etc. Whether there is an partial order which has all these but still not universal?

◮ If forbid long cycles, are the line graph interval orders still

universal?

◮ Density of line graph intervals orders.

What we do not know?

Thank you for your attention