Spanning trees of dense digraphs Tássio Naia

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Spanning trees of dense digraphs Tássio Naia

Joint work with

Richard Mycroft

ICGT 2018

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Spanning trees of dense digraphs Tássio Naia

Joint work with

Richard Mycroft

ICGT 2018

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Is there a copy of T in G?

tree T

v(T ) = n

tournament G

v(G) ≥ n

r(T)

r(T) := minimumk s.t. v(G)≥k =⇒ T ⊆G.

T is unavoidable : T ⊆G ∀G s.t. v(G)≥v(T).

So n≤r(T)≤2ⁿ⁻¹

T in every orientation of Kn ≡ T unavoidable ≡ r(T) =n

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Is there a copy of T in G?

tree T

v(T ) = n

tournament G

v(G) ≥ n

r(T)

So n≤r(T)≤2ⁿ⁻¹

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Is there a copy of T in G?

tree T

v(T ) = n

tournament G

v(G) ≥ n

r(T)

So n≤r(T)≤2ⁿ⁻¹

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Is there a copy of T in G?

tree T

v(T ) = n

tournament G

v(G) ≥ n

r(T) r(T) := minimumk s.t. v(G)≥k =⇒ T ⊆G.

T is unavoidable : T ⊆G ∀G s.t. v(G)≥v(T). So n≤r(T)≤2ⁿ⁻¹

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Is there a copy of T in G?

tree T

v(T ) = n

tournament G

v(G) ≥ n

r(T) r(T) := minimumk s.t. v(G)≥k =⇒ T ⊆G.

n−1

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Is there a copy of T in G?

tree T

v(T ) = n

tournament G

v(G) ≥ n

r(T)

r(T), Unavoidable tree r(T) := minimumk s.t. v(G)≥k =⇒ T ⊆G.

n−1

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Is there a copy of T in G?

tree T

v(T ) = n

tournament G

v(G) ≥ n

r(T)

n−1

≡ r(T) =n

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Is there a copy of T in G?

tree T

v(T ) = n

tournament G

v(G) ≥ n

r(T)

n−1

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Proof by picture

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Unavoidable trees — examples

Directed paths( Rédei ’34 ) · · ·

All large paths( Thomason ’86 )

All paths, 3 exceptions( Havet & Thomassé ’98 )

Some claws( Saks & Sós ’84; Lu ’93; Lu, Wang & Wong ’98 )

· · ·

≤

3

8 + 1 200

nbranches

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Unavoidable trees — examples

Directed paths( Rédei ’34 ) · · · All large paths( Thomason ’86 )

· · ·

≤

3

8 + 1 200

nbranches

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Unavoidable trees — examples

· · ·

≤

3

8 + 1 200

nbranches

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Unavoidable trees — examples

· · ·

≤

3

8 + 1 200

nbranches

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Avoidable trees — examples

· · · n−2

is not in n−3

And

5 vertices

is not in

3-regular

3-regular: 2·5−3 vertices

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Avoidable trees — examples

· · · n−2

is not in n−3

And

5 vertices

is not in

3-regular

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Avoidable trees — examples

· · · n−2

is not in n−3

And

5 vertices

is not in

3-regular

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Avoidable trees — examples

· · · n−2

is not in n−3

And

5 vertices

is not in

3-regular

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Avoidable trees — examples

· · · n−2

is not in n−3

And

5 vertices

is not in

3-regular

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Avoidable trees — examples

· · · n−2

is not in n−3

And

5 vertices

is not in

3-regular

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Avoidable trees — examples

· · · n−2

is not in n−3

And

5 vertices

is not in

3-regular

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Avoidable trees — examples

· · · n−2

is not in n−3

And

5 vertices

is not in

3-regular

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Avoidable trees — examples

· · · n−2

is not in n−3

And

5 vertices

is not in

3-regular

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Avoidable trees, a longstanding conjecture

Conjecture(Sumner, 1971)

? r(T)≤2n−2 for all T. v(T) =n

publ. who tournament size

1982 Chung n^1+o(n)

1983 Wormald nlog₂(2n/e)

1991 Häggkvist & Thomason 12nand also 4 +o(n)n

2002 Havet 38n/5−6

2000 Havet & Thomassé (7n−5)/2

2004 El Sahili 3n−3

2011 Kühn, Mycroft & Osthus 2n−2 for large n

Conjecture(Havet & Thomassé, 2002)

? T has`T leaves =⇒ r(T)≤n+`T −1. v(T) =n

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Avoidable trees, a longstanding conjecture

? r(T)≤2n−2 for all T. v(T) =n

1982 Chung n^1+o(n)

2002 Havet 38n/5−6

? T has`T leaves =⇒ r(T)≤n+`T −1. v(T) =n

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Avoidable trees, a longstanding conjecture

? r(T)≤2n−2 for all T. v(T) =n

1982 Chung n^1+o(n)

2002 Havet 38n/5−6

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Typical behaviour (1) — an approximate result

Kühn, Mycroft & Osthus, 2011 For allε,∆>0 and large n

∆(T)≤∆ =⇒ r(T)≤(1 +ε)n. v(T) =n

Mycroft, N., 2018 For allε, C >0 and large n

∆(T)≤(logn)^C =⇒ r(T)≤(1 +ε)n. v(T) =n

=⇒ (1 +ε)nvertices suffice to embed almost every tree

? IfT has`T leaves and v(G)≥n+`T −1 then T ⊆G.

Lemma Typical treeT has ≈n/e leaves and ∆(T)<logn.

Mycroft, N., 2018 Havet and Thomassé’s conjecture holds for almost everyT.

by Moon 1^st moment+Cayley

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Typical behaviour (1) — an approximate result

∆(T)≤∆ =⇒ r(T)≤(1 +ε)n. v(T) =n Mycroft, N., 2018 For allε, C >0 and large n

∆(T)≤(logn)^C =⇒ r(T)≤(1 +ε)n. v(T) =n

? IfT has`T leaves and v(G)≥n+`T −1 then T ⊆G.

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Typical behaviour (1) — an approximate result

∆(T)≤(logn)^C =⇒ r(T)≤(1 +ε)n. v(T) =n Moon, 1970 Typically, ∆(T) = (1±o(1))

logn

log logn

. T labelled

? IfT has`_T leaves and v(G)≥n+`_T −1 then T ⊆G.

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Typical behaviour (1) — an approximate result

∆(T)≤(logn)^C =⇒ r(T)≤(1 +ε)n. v(T) =n Moon, 1970 Typically, ∆(T) = (1±o(1))

logn

log logn

. T labelled

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Typical behaviour (1) — an approximate result

∆(T)≤(logn)^C =⇒ r(T)≤(1 +ε)n. v(T) =n

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Typical behaviour (1) — an approximate result

∆(T)≤(logn)^C =⇒ r(T)≤(1 +ε)n. v(T) =n

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Typical behaviour (1) — an approximate result

∆(T)≤(logn)^C =⇒ r(T)≤(1 +ε)n. v(T) =n

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Typical behaviour (1) — an approximate result

∆(T)≤(logn)^C =⇒ r(T)≤(1 +ε)n. v(T) =n

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Typical behaviour (1) — an approximate result

∆(T)≤(logn)^C =⇒ r(T)≤(1 +ε)n. v(T) =n

Mycroft, N., 2018 by Moon

1^st moment+Cayley

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Typical behaviour (2) — a probabilistic result

T_n:={n-vertex trees} oriented, labelled G_n:={n-vertex tournaments} labelled

Bender, Wormald, 1988 There exists

T ⊆ T_n with |T |= 1−o(1)T_n and G ⊆ G_n with |G|= 1−o(1)G_n s.t.

T ∈ T, G∈ G =⇒ T ⊆G.

Conjecture(Bender, Wormald, 1988)

? There existsT ⊆ T_n with |T |= 1−o(1)T_n s.t. T ∈ T =⇒ T unavoidable

Mycroft, N., 2018 Indeed!

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Typical behaviour (2) — a probabilistic result

T ∈ T, G∈ G =⇒ T ⊆G.

? There existsT ⊆ T_n with |T |= 1−o(1)T_n s.t. T ∈ T =⇒ T unavoidable

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Typical behaviour (2) — a probabilistic result

T ∈ T, G∈ G =⇒ T ⊆G.

? There existsT ⊆ T_n with |T |= 1−o(1)T_n s.t.

T ∈ T =⇒ T unavoidable

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Typical behaviour (2) — a probabilistic result

T ∈ T, G∈ G =⇒ T ⊆G.

? There existsT ⊆ T_n with |T |= 1−o(1)T_n s.t.

T ∈ T =⇒ T unavoidable

Mycroft, N., 2018

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Almost all trees are unavoidable!

Mycroft, N., 2018 There existsT_n⁰ ⊆ T_n of size 1−o(1)T_n s.t.

eachG∈ G_n contains all T ∈ T_n⁰ .

A sufficient condition which is typical !

Ingredients:

Mycroft, N., 2018

∆(T)≤polylog(n) =⇒ (1 +ε)n

Kühn, Mycroft & Osthus, 2011 Structure of large tournaments

+randomized embedding algorithm

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Almost all trees are unavoidable!

Mycroft, N., 2018 There existsT_n⁰ ⊆ T_n of size 1−o(1)T_n s.t.

eachG∈ G_n contains all T ∈ T_n⁰ .

A sufficient condition which is typical ! Ingredients:

Mycroft, N., 2018

∆(T)≤polylog(n) =⇒ (1 +ε)n

Kühn, Mycroft & Osthus, 2011 Structure of large tournaments

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Nice trees

··· ······ ······ ···· · · · · ·

pendant (out-star) in-leaf%

out-leaf%

α-nice

∀α >0, oriented tree T onnvertices is α-nice if

I αn pendant in-stars with an outleaf

I αn pendant out-stars with in- and outleaf

Mycroft, N., 2018 For allα, C >0and largen

T is α-nice and ∆(T)≤(logn)^C =⇒ T is unavoidable.

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Nice trees

··· ······ ······ ···· · · · · ·

out-leaf%

α-nice

∀α >0, oriented tree T onnvertices isα-nice if

T is α-nice and ∆(T)≤(logn)^C =⇒ T is unavoidable.

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Nice trees

··· ······ ······ ···· · · · · ·

out-leaf%

α-nice

∀α >0, oriented tree T onnvertices isα-nice if

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Nice trees are nice, and ALSO typical

T isα-nice and ∆(T)≤(logn)^C =⇒ T is unavoidable.

Moon, 1970 Almost alln-vertex trees have max. degree(1±o(1))

logn

log logn

.

Basic lemma Almost alln-vertex trees are ₂₅₀¹ -nice.

=⇒ Almost all trees are unavoidable.

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Nice trees are nice, and ALSO typical

logn

log logn

.

=⇒ Almost all trees are unavoidable.

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Nice trees are nice, and ALSO typical

logn

log logn

.

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Beyond tournaments — Dirac-type results

Komlós, Sárközy, Szemerédi, 1995

For all ∆>0, ε >0, and large n,

tree T, graphG s.t. n=v(T) =v(G) if ∆(T)≤∆ and δ(G)≥(1/2 +ε)n then T ⊆G.

Mycroft, N., 2018⁺ For all ∆>0, ε >0, and large n,

oriented tree T, digraph G s.t. n=v(T) =v(G) if ∆(T)≤∆ and δ⁰(G)≥(1/2 +ε)n

then T ⊆G. Work in progress:

1. Let ∆(T)grow withn

2. More general spanning subdigraphs (i.e.: allow some cycles)

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Beyond tournaments — Dirac-type results

Komlós, Sárközy, Szemerédi, 1995 For all ∆>0, ε >0, and large n,

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Beyond tournaments — Dirac-type results

tree T, graphG s.t. n=v(T) =v(G)

if ∆(T)≤∆ and δ(G)≥(1/2 +ε)n then T ⊆G.

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Beyond tournaments — Dirac-type results

tree T, graphG s.t. n=v(T) =v(G) if ∆(T)≤∆

and δ(G)≥(1/2 +ε)n then T ⊆G.

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Beyond tournaments — Dirac-type results

tree T, graphG s.t. n=v(T) =v(G) if ∆(T)≤∆ and δ(G)≥(1/2 +ε)n

then T ⊆G.

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Beyond tournaments — Dirac-type results

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Beyond tournaments — Dirac-type results

then T ⊆G.

Work in progress:

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Beyond tournaments — Dirac-type results

then T ⊆G.

Work in progress:

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Open questions

Q How about other oriented trees?

e.g., few leaves, or outbranchings

Q Characterise unavoidable trees

Q Orientations of n-chromatic graphs (Burr’s conjecture)

Conjecture (Burr, 1980)

? ^If^Gis a graph,χ(G) = 2n−2 andT is a tree on nvertices, then every orientation ofGcontains every orientation of T

Thanks! Other questions? Please ask!

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Open questions

Q How about other oriented trees?

e.g., few leaves, or outbranchings

Q Characterise unavoidable trees

Q Orientations of n-chromatic graphs (Burr’s conjecture)

Conjecture (Burr, 1980)

? ^If^Gis a graph,χ(G) = 2n−2 andT is a tree on nvertices, then every orientation ofGcontains every orientation of T

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Quick Reference

Paths and claws Avoidable trees Sumner

Bounded degree result

“Unavoidable” conjecture Result summary

Nice trees are nice — our sufficient condition

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Embedding T to G (general scheme)

I reserve a small set S ⊆G

I form T⁰ ⊆T by removing a few leaves

I embed T⁰ to G−S (using [MN18] ) I use S to cover badvertices

I use perfect matchings to complete the copy of T

T G

bad

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Embedding T to G (general scheme)

T

S G

bad

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Embedding T to G (general scheme)

T T⁰

S G

bad

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Embedding T to G (general scheme)

I embed T⁰ to G−S (using [MN18] )

I use S to cover badvertices

T T⁰

S G

bad

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Embedding T to G (general scheme)

T

S G

bad

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Embedding T to G (general scheme)

I use perfect matchings to complete the copy of T T

S G

bad

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Embedding T to G (general scheme)

I use perfect matchings to complete the copy of T T

S G

bad