Bandwidth, treewidth, separators, expansion, and universality fileBandwidth, treewidth, separators, expansion, and universality The bandwidth of planar graphs of bounded degree Bandwidth:

Bandwidth, treewidth, separators, expansion, and universality

Bandwidth, treewidth, separators, expansion, anduniversality

Julia Bottcher

Technische Universitat Munchen

Topological & Geometric Graph Theory, 2008

(joint work with Klaas P. Pruessmann, Anusch Taraz & Andreas Wurfl)

Julia Bottcher TU Munchen TGGT ’08


The bandwidth of planar graphs of bounded degreeBandwidth:

Let G be a graph on n vertices. Then bw(G) ≤ b if there is a labelling ofV (G) by 1, . . . , n s.t. for all {i, j} ∈ E(G) we have |i − j | ≤ b.

i j





i j

Examples:

Hamiltonian cycle: bandwidth 2





i j

Examples:


Star: bandwidth ⌊(n − 1)/2⌋





i j

Examples:


Star: bandwidth ⌊(n − 1)/2⌋Grid: bandwidth

√n





i j

Examples:



√n



The bandwidth of planar graphs of bounded degreeTheorem CHUNG’88

Let T be a tree on n vertices and maximal degree ∆(T ) ≤ ∆. Then T hasbandwidth at most O(n/ log∆ n).

Examples:



√n



The bandwidth of planar graphs of bounded degreeTheorem CHUNG’88

Let T be a tree on n vertices and maximal degree ∆(T ) ≤ ∆. Then T hasbandwidth at most O(n/ log∆ n).

Question: What about planar graphs?

Let ∆ be constant. Do planar graphs G on n vertices with ∆(G) ≤ ∆ havebandwidth o(n)?


√n



Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL

Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:

All n-vertex G ∈ G have bandwidth o(n),

All n-vertex G ∈ G have treewidth o(n),

All n-vertex G ∈ G have(

o(n), 2/3)

-separators,

All n-vertex G ∈ G are(

o(n), ε)

-bounded.








o(n), 2/3)

-separators,


o(n), ε)

-bounded.








o(n), 2/3)

-separators,


o(n), ε)

-bounded.








o(n), 2/3)

-separators,


o(n), ε)

-bounded.

tree decomposition








o(n), 2/3)

-separators,


o(n), ε)

-bounded.

width

tree decomposition








o(n), 2/3)

-separators,


o(n), ε)

-bounded.








o(n), 2/3)

-separators,


o(n), ε)

-bounded.








o(n), 2/3)

-separators,


o(n), ε)

-bounded.

≤ 23n

23n ≥ o(n)








o(n), 2/3)

-separators,


o(n), ε)

-bounded.

G is (b, ε)-bounded:

For all G ′ ⊆ G on n ′ ≥ b verticesthere is U ⊆ V (G ′) with |U| ≤ n ′

2such that |N(U)| ≤ ε|U|.

G








o(n), 2/3)

-separators,


o(n), ε)

-bounded.




G

G ′ ⊆ G








o(n), 2/3)

-separators,


o(n), ε)

-bounded.




G

G ′ ⊆ G

U N(U)








o(n), 2/3)

-separators,


o(n), ε)

-bounded.

Theorem LIPTON & TARJAN ’79

Planar graphs G on n vertices have (o(n), 2/3)-separators.








o(n), 2/3)

-separators,


o(n), ε)

-bounded.

Theorem ALON, SEYMOUR, THOMAS ’90

F -minor free G on n vertices have (|F |3/2o(n), 2/3)-separators.



Bandwidth versus path partition width

Path partition

A partition V (G) = V1 _∪ . . . _∪Vt of G such that edges of G only run withinthe Vi and between Vi and Vi+1.The width of this partition is maxi |Vi |, and ppw(G) is the minimal width ofa path partition of G.




Path partition


V1 V2 Vt




Path partition


V1 V2 Vt

Observation

For all G we have ppw(G) ≤ bw(G) ≤ 2 ppw(G).



The proof

All n-vertex G = (V , E) ∈ G have(

o(n), 2/3)

-separators ⇒

All n-vertex G ∈ G have bandwidth o(n).



The proof


o(n), 2/3)

-separators ⇒


Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).



The proof


o(n), 2/3)

-separators ⇒



G



The proof


o(n), 2/3)

-separators ⇒



G



The proof


o(n), 2/3)

-separators ⇒



G



The proof


o(n), 2/3)

-separators ⇒



G

S

R1 R2

Rt



The proof


o(n), 2/3)

-separators ⇒



G

S

R1 R2

Rt

Bi := {v : v ∈ Rj , i < j,dist(v , S) = i}



The proof


o(n), 2/3)

-separators ⇒



G

S

R ′

1 R ′

2

R ′

t

B Bi := {v : v ∈ Rj , i < j,dist(v , S) = i}

B :=⋃

Bi , R ′

j := Rj \ B



The proof


o(n), 2/3)

-separators ⇒



G

S

R ′

1 R ′

2

R ′

t


B :=⋃

Bi , R ′

j := Rj \ B

|S|, |R ′

j | = o(n)

⇒ |Bi | ≤ ∆t |S| = o(n)



The proof


o(n), 2/3)

-separators ⇒



G

S

R ′

1 R ′

2

R ′

t


B :=⋃

Bi , R ′

j := Rj \ B

|S|, |R ′

j | = o(n)

⇒ |Bi | ≤ ∆t |S| = o(n)

V1 := S



The proof


o(n), 2/3)

-separators ⇒



G

S

R ′

1 R ′

2

R ′

t


B :=⋃

Bi , R ′

j := Rj \ B

|S|, |R ′

j | = o(n)

⇒ |Bi | ≤ ∆t |S| = o(n)

V1 := S

Vi+1 := R ′

i _∪Bi



A consequence: universality for planar graphs

Theorem

Every n-vertex graph H with min. degree δ(H) ≥ (34 + γ)n contains every

n-vertex planar graph G with max. degree ∆.




Theorem

∀γ > 0, ∆∃n0 ∀n ≥ n0






Theorem

∀γ > 0, ∆∃n0 ∀n ≥ n0



Universality for Graphs of Bounded Bandwidth B,SCHACHT,TARAZ’08

For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t. for all n ≥ n0

χ(G) = k , ∆(G) ≤ ∆, bw(G) ≤ βn

δ(H) ≥ ( k−1k + γ)n =⇒ H contains G




Theorem

∀γ > 0, ∆∃n0 ∀n ≥ n0





χ(G) = k , ∆(G) ≤ ∆, bw(G) ≤ βn


Theorem KUHN, OSTHUS, TARAZ 2005

Every n-vertex graph H with min. degree δ(H) ≥ (23 + γ)n contains some

n-vertex planar triangulation G.




Theorem

∀γ > 0, ∆∃n0 ∀n ≥ n0


n-vertex planar graph G with max. degree ∆ and chromatic number 3.



χ(G) = k , ∆(G) ≤ ∆, bw(G) ≤ βn


Theorem KUHN, OSTHUS, TARAZ 2005

Every n-vertex graph H with min. degree δ(H) ≥ (23 + γ)n contains some

n-vertex planar triangulation G.



Concluding remarks

Let G be a hereditary class of graphs with max. degree ∆. Then

Bandwidth, treewidth, separators, expansion are sublinearlyequivalent for G.

Each of them implies that all H with δ(H) ≥ ( r−1r + γ)n are universal

for the class of r -chromatic graphs in G.



Concluding remarks





Question

Let G be an r -chromatic expander on n vertices (i.e. ∀U ⊆ V (G) with|U| ≤ n

2 we have |N(U)| ≥ εU for some constant ε > 0).Is it true that there is an n-vertex H with δ(H) ≥ ( r−1

r + γ)n but G 6⊆ H forγ > 0 sufficiently small?



Concluding remarks





Question

Let G be an r -chromatic expander on n vertices (i.e. ∀U ⊆ V (G) with|U| ≤ n

2 we have |N(U)| ≥ εU for some constant ε > 0).Is it true that there is an n-vertex H with δ(H) ≥ ( r−1

r + γ)n but G 6⊆ H forγ > 0 sufficiently small?

Merci beaucoup.Julia Bottcher TU Munchen TGGT ’08

Bandwidth, treewidth, separators, expansion, and universality fileBandwidth, treewidth, separators, expansion, and universality The bandwidth of planar graphs of bounded degree Bandwidth:

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