Unavoidable trees in tournaments Richard Mycroft Tássio Naia 20 April 2016 1
Tournaments & Oriented Trees Oriented tree T on n vertices, tournament G 2
Tournaments & Oriented Trees Oriented tree T on n vertices, tournament G Is there a copy of T in G ? | V ( T ) | = n ≤ | V ( G ) | 2
Tournaments & Oriented Trees Oriented tree T on n vertices, tournament G Is there a copy of T in G ? | V ( T ) | = n ≤ | V ( G ) | 2
Tournaments & Oriented Trees Oriented tree T on n vertices, tournament G Is there a copy of T in G ? | V ( T ) | = n ≤ | V ( G ) | Definition (unavoidable trees) A (oriented) tree T with | V ( T ) | = n is unavoidable if every tournament on n vertices contains a copy of T . 2
Unavoidable trees — examples Directed paths ( Rédei 1934 ) · · · 3
Unavoidable trees — examples Directed paths ( Rédei 1934 ) · · · All large paths ( Thomason ’86 ) 3
Unavoidable trees — examples Directed paths ( Rédei 1934 ) · · · All large paths ( Thomason ’86 ) All paths, 3 exceptions ( Havet & Thomassé ’98 ) 3
Unavoidable trees — examples Directed paths ( Rédei 1934 ) · · · All large paths ( Thomason ’86 ) All paths, 3 exceptions ( Havet & Thomassé ’98 ) Some claws ( Saks & Sós 84; Lu ’93; Lu, Wang & Wong ’98 ) · · · · · · � 3 1 � ≤ 8 + n branches 200 · · · · · · 3
Examples — non-unavoidable trees · · · n − 2 4
Examples — non-unavoidable trees · · · is not in n − 3 n − 2 4
Examples — non-unavoidable trees · · · is not in n − 3 n − 2 And 5 vertices 4
Examples — non-unavoidable trees · · · is not in n − 3 n − 2 And is not in 5 vertices 3-regular 4
Examples — non-unavoidable trees · · · is not in n − 3 n − 2 And is not in 5 vertices 3-regular 4
Examples — non-unavoidable trees · · · is not in n − 3 n − 2 And is not in 5 vertices 3-regular 4
Examples — non-unavoidable trees · · · is not in n − 3 n − 2 And is not in 5 vertices 3-regular 4
Examples — non-unavoidable trees · · · is not in n − 3 n − 2 And is not in 5 vertices 3-regular: 2 · 5 − 3 vertices 4
Conjecture and proofs Sumner’s conjecture (1971) Every oriented tree on n vertices is contained in every tournament on 2 n − 2 vertices. 5
Conjecture and proofs Sumner’s conjecture (1971) Every oriented tree on n vertices is contained in every tournament on 2 n − 2 vertices. publ. who tournament size n 1+o( n ) 1982 Chung 1983 Wormald n log 2 (2 n / e ) � 4 + o( n ) � n 1991 Häggkvist & Thomason 12 n and also 2002 Havet 38 n / 5 − 6 2000 Havet & Thomassé (7 n − 5) / 2 2004 El Sahili 3 n − 3 2011 Kühn, Mycroft & Osthus 2 n − 2 for large n 5
Embedding bounded-degree trees Theorem (Kühn, Mycroft & Osthus, 2011) For all α, ∆ > 0 there exists n 0 such that if n > n 0 , each tournament on (1 + α ) n vertices contains any tree T on n vertices with ∆( T ) ≤ ∆ . 6
When can we do better? Question (Alon) Which trees are unavoidable? 7
When can we do better? Question (Alon) Which trees are unavoidable? Paths, 7
When can we do better? Question (Alon) Which trees are unavoidable? Paths, some claws , . . . . . . . . . . . . � 3 1 � ≤ 8 + n branches 200 7
When can we do better? Question (Alon) Which trees are unavoidable? Paths, some claws , this tree: 7 vertices . . . . . . . . . . . . � 3 1 � ≤ 8 + n branches 200 7
A family of examples – alternating trees Alternating trees are rooted trees B ℓ B 1 : r ( B 1 ) 8
A family of examples – alternating trees Alternating trees are rooted trees B ℓ r ( B i +1 ) B 1 : B i +1 : r ( B i ) r ( B i ) r ( B 1 ) B i B i 8
A family of examples – alternating trees Alternating trees are rooted trees B ℓ r ( B i +1 ) B 1 : B i +1 : r ( B i ) r ( B i ) r ( B 1 ) B i B i B 1 , B 2 and B 3 are unavoidable: 8
A family of examples – alternating trees Alternating trees are rooted trees B ℓ r ( B i +1 ) B 1 : B i +1 : r ( B i ) r ( B i ) r ( B 1 ) B i B i B 1 , B 2 and B 3 are unavoidable: Theorem (Mycroft, N. 2016 + ) For ℓ large enough, B ℓ is unavoidable. 8
More examples – balanced q -ary trees q -ary tree are rooted trees B q q ∈ N ℓ B q 1 : r ( B q 1 ) 9
More examples – balanced q -ary trees q -ary tree are rooted trees B q q ∈ N ℓ r ( B q i +1 ) B q B q 1 : i +1 : r ( B q 1 ) r ( B q r ( B q · · · r ( B q i ) i ) i ) B q B q B q i i i q copies 9
More examples – balanced q -ary trees q -ary tree are rooted trees B q q ∈ N ℓ r ( B q i +1 ) B q B q 1 : i +1 : r ( B q 1 ) r ( B q r ( B q · · · r ( B q i ) i ) i ) B q B q B q i i i q copies Theorem (Mycroft, N. 2016 + ) For each q ∈ N , if ℓ large enough then almost all orientations of B q ℓ are unavoidable. 9
More examples – balanced q -ary trees q -ary tree are rooted trees B q q ∈ N ℓ r ( B q i +1 ) B q B q 1 : i +1 : r ( B q 1 ) r ( B q r ( B q · · · r ( B q i ) i ) i ) B q B q B q i i i q copies Theorem (Mycroft, N. 2016 + ) For each q ∈ N , if ℓ large enough then almost all orientations of B q ℓ are unavoidable. The method works a much wider class of trees. 9
Some definitions and a property of B ℓ centre B 2 is a cherry: 10
Some definitions and a property of B ℓ centre B 2 is a cherry: in-leaf 10
Some definitions and a property of B ℓ centre B 2 is a cherry: in-leaf out-leaf 10
Some definitions and a property of B ℓ centre B 2 is a cherry: in-leaf out-leaf B ℓ has many pendant cherries 10
Some definitions and a property of B ℓ centre B 2 is a cherry: in-leaf out-leaf B ℓ has many pendant cherries out cherry in cherry 10
Characterization of large tournaments Theorem (Kühn, Mycroft, Osthus 2011) Large tournaments contain either a large strong cut or a large robust expander of linear minimum semidegree. 11
Characterization of large tournaments Theorem (Kühn, Mycroft, Osthus 2011) Large tournaments contain either a large strong cut or a large robust expander of linear minimum semidegree. L R bad 11
Characterization of large tournaments Theorem (Kühn, Mycroft, Osthus 2011) Large tournaments contain either a large strong cut or a large robust expander of linear minimum semidegree. robust expander or L R of linear semidegree bad 11
Characterization of large tournaments Theorem (Kühn, Mycroft, Osthus 2011) Large tournaments contain either a large strong cut or a large robust expander of linear minimum semidegree. Theorem (Kühn, Osthus, Treglown 2010) A large robust expander of linear minimum semidegree contains a regular cycle of cluster tournaments. or L R bad 11
Embedding B ℓ to G (general scheme) B ℓ G 12
Embedding B ℓ to G (general scheme) ◮ reserve a small set S ⊆ G B ℓ G S 12
Embedding B ℓ to G (general scheme) ◮ reserve a small set S ⊆ G ◮ form T ′ ⊆ B ℓ removing a few leaves B ℓ G T ′ S 12
Embedding B ℓ to G (general scheme) ◮ reserve a small set S ⊆ G ◮ form T ′ ⊆ B ℓ removing a few leaves ◮ embed T ′ to G − S ( uses [KMO ’11] ) B ℓ G T ′ S 12
Embedding B ℓ to G (general scheme) ◮ reserve a small set S ⊆ G ◮ form T ′ ⊆ B ℓ removing a few leaves ◮ embed T ′ to G − S ( uses [KMO ’11] ) ◮ use S to cover tricky vertices B ℓ G 12
Embedding B ℓ to G (general scheme) ◮ reserve a small set S ⊆ G ◮ form T ′ ⊆ B ℓ removing a few leaves ◮ embed T ′ to G − S ( uses [KMO ’11] ) ◮ use S to cover tricky vertices ◮ use perfect matchings to complete the copy of B ℓ B ℓ G 12
Embedding B ℓ to G (general scheme) ◮ reserve a small set S ⊆ G ◮ form T ′ ⊆ B ℓ removing a few leaves ◮ embed T ′ to G − S ( uses [KMO ’11] ) ◮ use S to cover tricky vertices ◮ use perfect matchings to complete the copy of B ℓ B ℓ G 12
Beyond binary trees Theorem (R. Mycroft, N., 2016 + ) For all q > 0 there exists n 0 such that if n > n 0 almost all orientations of every “roughly balanced” q-ary tree on n vertices are unavoidable. 13
Beyond binary trees Theorem (R. Mycroft, N., 2016 + ) For all q > 0 there exists n 0 such that if n > n 0 almost all orientations of every “roughly balanced” q-ary tree on n vertices are unavoidable. Work in progress For all ∆ > 0 there exists n 0 such that for n > n 0 almost all labelled trees T on n vertices with ∆( T ) ≤ ∆ are unavoidable. 13
Beyond binary trees Theorem (R. Mycroft, N., 2016 + ) For all q > 0 there exists n 0 such that if n > n 0 almost all orientations of every “roughly balanced” q-ary tree on n vertices are unavoidable. Work in progress For all ∆ > 0 there exists n 0 such that for n > n 0 almost all labelled trees T on n vertices with ∆( T ) ≤ ∆ are unavoidable. ◮ most labelled undirected trees have pendant cherries ◮ most orientations of a labelled tree have good cherry orientations 13
Recommend
More recommend
