Let f ( a ) be the minimum integer such that every graph of average - PowerPoint PPT Presentation

Let f ( a ) be the minimum integer such that every graph of average degree at least f ( a ) contains K a as a minor. Theorem (Thomasson) f ( a ) = ( 0 . 638 . . . + o ( 1 )) a log a Theorem (Norin and Thomas) For every a there exists N such that every a-connected graph G with at least N vertices either contains K a as a minor, or is obtained from a planar graph by adding at most a − 5 apex vertices.

We will prove a simpler result: Theorem (Böhme, Kawarabayashi, Maharry and Mohar) Every ( 3 a + 2 ) -connected graph of minimum degree at least 20 a and with ≫ a , k , s , t vertices either contains sK a , k as a minor, or contains a subdivision of K a , t .

Observation K a � K a − 1 , a Corollary Every ( 3 a − 1 ) -connected graph of minimum degree at least 20 a and with ≫ a vertices contains K a as a minor. Corollary Every ( 3 a + 2 ) -connected graph of minimum degree at least 20 a, maximum degree less than t, and with ≫ a , k , s , t vertices contains sK a , k as a minor.

Definition A graph M is k -linked if for all distinct s 1 , . . . , s k , t 1 , . . . , t k ∈ V ( G ) , M contains disjoint paths from s 1 to t 1 , . . . , s k to t k . Theorem A graph of average degree at least 13 k contains a k-linked subgraph.

A path decomposition ( x 0 x 1 . . . x m , β ) of H . S i = β ( x i − 1 ) ∩ β ( x i ) . Definition q -linked if | S 1 | = | S 2 | = . . . = | S m | = q and H contains q vertex-disjoint linking paths from S 1 to S m .

A vertex v is internal if it does not belong to β ( x 0 ) ∪ β ( x m ) ∪ � m i = 1 S i . i v : v ∈ β ( x i v ) β v = β ( x i v ) , L ( v ) = S i v − 1 , R v = S i v . Focus F : Set of internal vertices belonging to distinct bags.

A linking path P is F -universal if there exists u P ∈ V ( P ) such that V ( P ) ∩ β v = { u P } for all v ∈ F . F -transversal if V ( P ) ∩ β v and V ( P ) ∩ β v ′ are disjoint for all distinct v , v ′ ∈ F . Observation If | F | ≥ ( ℓ + 3 ) ℓ , then there exists F ′ ⊆ F of size at least ℓ such that P is either F ′ -universal or F ′ -transversal.

For v ∈ F , let Γ v be the graph with V (Γ v ) = { P 1 , . . . , P q } , P i P j ∈ E (Γ v ) iff H [ β v ] contains a path from P i to P j disjoint from all other linking paths. Observation If | F | ≫ ℓ , then there exists F ′ ⊆ F of size at least ℓ such that Γ v = Γ v ′ for all v , v ′ ∈ F ′ .

Lemma Assume for every v ∈ F: v lies on P 1 . No separation ( A , B ) of H [ β v ] of order less than 3 a + 2 with L v ∪ R v ∪ { v } ⊆ A and B �⊆ A. Vertices of β v \ ( L v ∪ R v ) have degree at least 20 a − 4 in H [ β v ] . If | F | ≫ a , k , s , t , q, then H contains sK a , k as a minor or K a , t as a topological minor. Assume uniformity; U : F -universal paths (vertices). Γ = Γ v for v ∈ F . Γ 1 : the U -bridge containing P 1 , Γ 0 = Γ 1 − U .

For each v ∈ F , let H 1 ( v ) be the U -bridge of H [ β v ] containing v . H 1 ( v ) is intersected exactly by linking paths in Γ 1 . Let H 1 consist of linking paths in Γ 1 and H 1 ( v ) for v ∈ F . Let H 0 ( v ) = H 1 ( v ) − U , H 0 = H 1 − U .

Observation � q � If for sk vertices v ∈ F, there exists x ∈ V ( H 0 ( v )) with a ≥ 2 a + 1 neighbors in ( L v ∪ R v ) \ U, then sK a , k � H.

Observation � q � If for t vertices v ∈ F, there exist a disjoint paths from v to U a in H 1 ( v ) , then H contains a subdivision of K a , t .

For all other v ∈ F : Separation ( A v , B v ) of H 1 ( v ) of order less than a with v ∈ V ( A v ) \ V ( B v ) and U ∩ V ( H 1 ) ⊆ V ( B v ) . H [ A v − ( { v } ∪ L v ∪ R v ∪ V ( B v ))] has minimum degree at least 17 a − 5 ⇒ ( a + 1 ) -linked subgraph M v . By assumptions: 3 a + 2 disjoint paths from M v to L v ∪ R v ∪ { v } . 2 a + 2 end in ( L v ∪ R v ) \ U . Can be redirected so that a + 1 end in X v ⊆ L v \ U and a + 1 in Y v ⊆ R v \ U .

For all other v ∈ F : Separation ( A v , B v ) of H 1 ( v ) of order less than a with v ∈ V ( A v ) \ V ( B v ) and U ∩ V ( H 1 ) ⊆ V ( B v ) . H [ A v − ( { v } ∪ L v ∪ R v ∪ V ( B v ))] has minimum degree at least 17 a − 5 ⇒ ( a + 1 ) -linked subgraph M v . By assumptions: 3 a + 2 disjoint paths from M v to L v ∪ R v ∪ { v } . 2 a + 2 end in ( L v ∪ R v ) \ U . Can be redirected so that a + 1 end in X v ⊆ L v \ U and a + 1 in Y v ⊆ R v \ U .

For all other v ∈ F : Separation ( A v , B v ) of H 1 ( v ) of order less than a with v ∈ V ( A v ) \ V ( B v ) and U ∩ V ( H 1 ) ⊆ V ( B v ) . H [ A v − ( { v } ∪ L v ∪ R v ∪ V ( B v ))] has minimum degree at least 17 a − 5 ⇒ ( a + 1 ) -linked subgraph M v . By assumptions: 3 a + 2 disjoint paths from M v to L v ∪ R v ∪ { v } . 2 a + 2 end in ( L v ∪ R v ) \ U . Can be redirected so that a + 1 end in X v ⊆ L v \ U and a + 1 in Y v ⊆ R v \ U .

Observation If there are at least a + 1 vertices of F between v and v ′ and A ⊆ R v and B ⊆ L v ′ have size a + 1 , then H 0 contains a + 1 disjoint paths from A to B.

Observation If there are at least a + 1 vertices of F between v and v ′ and A ⊆ R v and B ⊆ L v ′ have size a + 1 , then H 0 contains a + 1 disjoint paths from A to B.

sK a , k � H .

A tree decomposition ( T , β ) of a graph G is linked if for any x , y ∈ V ( T ) and an integer k , either G contains k vertex-disjoint paths from β ( x ) to β ( y ) , or there exists z ∈ V ( T ) separating x from y in T such that | β ( z ) | < k . nondegenerate if no two bags are the same. Theorem (Thomas) Every graph G has a nondegenerate linked tree decomposition of width tw ( G ) .

Lemma Every ( 3 a + 2 ) -connected graph of minimum degree at least 20 a, treewidth at most q and with ≫ a , k , s , t , q vertices either contains sK a , k as a minor or K a , t as a topological minor.

Lemma Every ( 3 a + 2 ) -connected graph of minimum degree at least 20 a, treewidth at most q and with ≫ a , k , s , t , q vertices either contains sK a , k as a minor or K a , t as a topological minor.

Local structure decomposition with a wall:

Apex vertices A , a vortex R with boundary ∂ R = v 0 v 1 . . . v m . R is q -linked: its path decomposition ( v 1 . . . v m , β ) satisfies β ( v i ) ∩ { v 0 , . . . , v m } = { v i − 1 , v i } for i = 1 , . . . , m , | β ( v i ) ∩ β ( v i + 1 ) | = q + 1 for i = 1 , . . . , m − 1, and R − ∂ R contains q paths from β ( v 1 ) to β ( v m ) . v ∈ ∂ R is local if all but at most four neighbors of v belong to V ( A ∪ R ) . F ⊆ ∂ R is attached to a comb if there exist paths outside of R ∪ A from F to a path, ending in order.

Lemma ( 3 a + 2 ) -connected graph G of minimum degree at least 20 a, apices A, q-linked vortex R, F ⊆ ∂ R local vertices attached to a comb, | F | ≫ a , k , s , t , q , | A | ⇒ G contains sK a , k as a minor or K a , t as a topological minor.

Many vertices with ≥ a neighbors in A , or many pieces attaching to the surface part imply subdivision of K a , t .

Large treewidth ⇒ large wall W . No sK a , k -minor ⇒ decomposition with respect to W ; we can assume the vortices are linked. No subdivision of K a , t ⇒ subwall W ′ with no attaching parts, all vertices < a neighbors in A , | V ( W ′ ) | ≥ M . Local vertices of vortices cut off by small cuts → G ′ . Contract vortex interiors ⇒ ≤ M vertices of degree < 6. 6 ( | V ( G ′ ) | − M ) + ( 19 a − 6 ) M ≤ 2 | E ( G ′ ) | ≤ 6 | V ( G ′ ) | + 6 g �