Universality properties of random graphs Rajko Nenadov joint work - PowerPoint PPT Presentation

Universality properties of random graphs Rajko Nenadov joint work with David Conlon, Asaf Ferber and Nemanja ˇ Skori´ c

Embedding – definition Given graphs G and H , an injective function f : V ( H ) → V ( G ) is an embedding of H into G if { v, u } ∈ E ( H ) ⇒ { f ( v ) , f ( u ) } ∈ E ( G )

Embedding – definition Given graphs G and H , an injective function f : V ( H ) → V ( G ) is an embedding of H into G if { v, u } ∈ E ( H ) ⇒ { f ( v ) , f ( u ) } ∈ E ( G )

Embedding – definition Given graphs G and H , an injective function f : V ( H ) → V ( G ) is an embedding of H into G if { v, u } ∈ E ( H ) ⇒ { f ( v ) , f ( u ) } ∈ E ( G )

Embedding – definition Given graphs G and H , an injective function f : V ( H ) → V ( G ) is an embedding of H into G if { v, u } ∈ E ( H ) ⇒ { f ( v ) , f ( u ) } ∈ E ( G ) Not necessarily induced!

Random graphs Binomial random graph G ( n, p ) graph on n vertices each edge present with probability p (independently)

Random graphs Binomial random graph G ( n, p ) graph on n vertices each edge present with probability p (independently) Theorem (Bollob´ as, Thomason ’87) – threshold functions For every monotone graph property P (connected, Hamiltonian, etc.) there exists p 0 = p 0 ( n ) such that ( 1 , p � p 0 ( n ) n →∞ Pr[ G ( n, p ) 2 P ] = lim 0 , p ⌧ p 0 ( n ) .

Embeddings in random graphs Binomial random graph G ( n, p ) graph on n vertices each edge present with probability p (independently) Given a sequence of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ H n ⊂ G ( n, p )] = 1? lim

Embeddings in random graphs Binomial random graph G ( n, p ) graph on n vertices each edge present with probability p (independently) Given a sequence of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ H n ⊂ G ( n, p )] = 1? lim In this talk we are interested in the case when H n satisfies the following: (i) v ( H n ) ≤ (1 − ε ) n (”almost-spanning”) (ii) ∆ ( H n ) ≤ ∆ (”bounded-degree”)

Embeddings in random graphs Given a sequence of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ H n ⇢ G ( n, p )] = 1? lim Theorem (Alon, F¨ uredi ’91) – constructive proof If H n has maximum degree at most ∆ , then ◆ 1 / ∆ ✓ log n p � n su ffi ces. (Even for ε = 0 )

Embeddings in random graphs Given a sequence of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ H n ⇢ G ( n, p )] = 1? lim Theorem (Alon, F¨ uredi ’91) – constructive proof If H n has maximum degree at most ∆ , then ◆ 1 / ∆ ✓ log n p � n su ffi ces. (Even for ε = 0 ) Better bounds obtained by Riordan using the second-moment method; non-constructive!

Universality Given a sequence of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ H n ⊂ G ( n, p )] = 1? lim

Universality Given a sequence of families of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ for every graph H n ∈ H n : H n ⊂ G ( n, p )] = 1? lim

Universality Given a sequence of families of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ for every graph H n ∈ H n : H n ⊂ G ( n, p ) lim ] = 1? | {z } G ( n, p ) is H n -universal For which p does G ( n, p ) simultaneously contain every H n ∈ H n ?

Universality Given a sequence of families of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ for every graph H n ∈ H n : H n ⊂ G ( n, p ) lim ] = 1? | {z } G ( n, p ) is H n -universal For which p does G ( n, p ) simultaneously contain every H n ∈ H n ? In this talk H n ( ε , ∆ ) = { all almost-spanning bounded-degree graphs } = { H n : v ( H n ) ≤ (1 − ε ) n and ∆ ( H n ) ≤ ∆ }

6 Universality Given a sequence of families of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ for every graph H n 2 H n : H n ⇢ G ( n, p ) lim ] = 1? | {z } G ( n, p ) is H n -universal Note lim n →∞ Pr[ H n ⇢ G ( n, p )] = 1 for a sequence of graphs H n 2 H n ) = lim n →∞ [ G ( n, p ) is H n -universal ] = 1

6 Universality Given a sequence of families of graphs ( H n ) n →∞ , for which p = p ( n ) we have n →∞ Pr[ for every graph H n 2 H n : H n ⇢ G ( n, p ) lim ] = 1? | {z } G ( n, p ) is H n -universal Note lim n →∞ Pr[ H n ⇢ G ( n, p )] = 1 for a sequence of graphs H n 2 H n ) = lim n →∞ [ G ( n, p ) is H n -universal ] = 1 useless if H is large z }| { X Pr[ G ( n, p ) is not H n -universal ]  Pr[ H n 6⇢ G ( n, p )] H ∈ H n

Universality in random graphs Alon, Capalbo, Kohayakawa, R¨ odl, Ruci´ nski and Szemer´ edi ’00: Theorem For any constant ∆ 2 N and ε > 0 , if ◆ 1 / ∆ ✓ log n p � n then G ( n, p ) is a.a.s. H n ( ε , ∆ ) -universal. (a.a.s = asymptotically almost surely, i.e. with probability tending to 1 as n ! 1 )

Universality in random graphs Alon, Capalbo, Kohayakawa, R¨ odl, Ruci´ nski and Szemer´ edi ’00: Theorem For any constant ∆ 2 N and ε > 0 , if ◆ 1 / ∆ ✓ log n p � n then G ( n, p ) is a.a.s. H n ( ε , ∆ ) -universal. (a.a.s = asymptotically almost surely, i.e. with probability tending to 1 as n ! 1 ) Remark: improved to ε = 0 (spanning) by Dellamonica, Kohayakawa, R¨ odl and Ruci´ nski (’12) and Kim and Lee (’15)

� 1 / ∆ ⇣ log n A story about n

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k .

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!!

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set

A story about (log n/n ) 1 / ∆ Fact � 1 / ∆ ⇣ log n If p � then G ( n, p ) a.a.s. has the property that every n set of k  ∆ vertices has a common neighborhood of size ⇡ np k . Importantly , it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set All previous results in some way implement this approach.

Our result Theorem [ACKRRSz ’00] For any constant ∆ 2 N and ε > 0 , if ◆ 1 ✓ log n ∆ p � n then G ( n, p ) is a.a.s. H n ( ε , ∆ ) -universal.

Our result Theorem [Conlon, Ferber, N., ˇ Skori´ c ’16] For any constant ∆ 2 N ( ∆ � 3 ) and ε > 0 , if 1 ✓ log 3 n ◆ ∆ − 1 p � n then G ( n, p ) is a.a.s. H n ( ε , ∆ ) -universal.