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 Skori c Embedding definition Given graphs G and H , an injective function f : V ( H ) V ( G ) is an embedding of H into G
Rajko Nenadov
joint work with David Conlon, Asaf Ferber and Nemanja ˇ Skori´ c
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)
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)
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)
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!
Binomial random graph G(n, p) graph on n vertices each edge present with probability p (independently)
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 p0 = p0(n) such that lim
n→∞ Pr[G(n, p) 2 P] =
( 1, p p0(n) 0, p ⌧ p0(n).
Binomial random graph G(n, p) graph on n vertices each edge present with probability p (independently) Given a sequence of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[Hn ⊂ G(n, p)] = 1?
Binomial random graph G(n, p) graph on n vertices each edge present with probability p (independently) Given a sequence of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[Hn ⊂ G(n, p)] = 1?
In this talk we are interested in the case when Hn satisfies the following: (i) v(Hn) ≤ (1 − ε)n (”almost-spanning”) (ii) ∆(Hn) ≤ ∆ (”bounded-degree”)
Given a sequence of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[Hn ⇢ G(n, p)] = 1?
Theorem (Alon, F¨ uredi ’91) – constructive proof If Hn has maximum degree at most ∆, then p ✓log n n ◆1/∆
Given a sequence of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[Hn ⇢ G(n, p)] = 1?
Theorem (Alon, F¨ uredi ’91) – constructive proof If Hn has maximum degree at most ∆, then p ✓log n n ◆1/∆
Better bounds obtained by Riordan using the second-moment method; non-constructive!
Given a sequence of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[Hn ⊂ G(n, p)] = 1?
Given a sequence of families of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[for every graph Hn ∈ Hn : Hn ⊂ G(n, p)] = 1?
Given a sequence of families of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[for every graph Hn ∈ Hn : Hn ⊂ G(n, p)
| {z }
] = 1? For which p does G(n, p) simultaneously contain every Hn ∈ Hn?
Given a sequence of families of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[for every graph Hn ∈ Hn : Hn ⊂ G(n, p)
| {z }
] = 1? For which p does G(n, p) simultaneously contain every Hn ∈ Hn? In this talk Hn(ε, ∆) = { all almost-spanning bounded-degree graphs } = {Hn : v(Hn) ≤ (1 − ε)n and ∆(Hn) ≤ ∆}
Given a sequence of families of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[for every graph Hn 2 Hn : Hn ⇢ G(n, p)
| {z }
] = 1? Note limn→∞ Pr[Hn ⇢ G(n, p)] = 1 for a sequence of graphs Hn 2 Hn
limn→∞[G(n, p) is Hn-universal] = 1
Given a sequence of families of graphs (Hn)n→∞, for which p = p(n) we have lim
n→∞ Pr[for every graph Hn 2 Hn : Hn ⇢ G(n, p)
| {z }
] = 1? Note limn→∞ Pr[Hn ⇢ G(n, p)] = 1 for a sequence of graphs Hn 2 Hn
limn→∞[G(n, p) is Hn-universal] = 1 Pr[G(n, p) is not Hn-universal]
useless if H is large
z }| { X
H∈Hn
Pr[Hn 6⇢ G(n, p)]
Alon, Capalbo, Kohayakawa, R¨
nski and Szemer´ edi ’00: Theorem For any constant ∆ 2 N and ε > 0, if p ✓log n n ◆1/∆ then G(n, p) is a.a.s. Hn(ε, ∆)-universal. (a.a.s = asymptotically almost surely, i.e. with probability tending to 1 as n ! 1)
Alon, Capalbo, Kohayakawa, R¨
nski and Szemer´ edi ’00: Theorem For any constant ∆ 2 N and ε > 0, if p ✓log n n ◆1/∆ then G(n, p) is a.a.s. Hn(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¨
nski (’12) and Kim and Lee (’15)
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk.
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!!
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. 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.
Theorem [ACKRRSz ’00] For any constant ∆ 2 N and ε > 0, if p ✓log n n ◆ 1
∆
then G(n, p) is a.a.s. Hn(ε, ∆)-universal.
Theorem [Conlon, Ferber, N., ˇ Skori´ c ’16] For any constant ∆ 2 N (∆ 3) and ε > 0, if p ✓log3 n n ◆
1 ∆−1
then G(n, p) is a.a.s. Hn(ε, ∆)-universal.
Theorem [Conlon, Ferber, N., ˇ Skori´ c ’16] For any constant ∆ 2 N (∆ 3) and ε > 0, if p ✓log3 n n ◆
1 ∆−1
then G(n, p) is a.a.s. Hn(ε, ∆)-universal. Remark This is optimal (up to the logarithmic factor) for ∆ = 3: consider a disjoint union of (1−ε)n
4
copies of K4
Theorem [Conlon, Ferber, N., ˇ Skori´ c ’16] For any constant ∆ 2 N (∆ 3) and ε > 0, if p ✓log3 n n ◆
1 ∆−1
then G(n, p) is a.a.s. Hn(ε, ∆)-universal. Theorem [ACKRRSz ’00] If p ✓log n n ◆ 1
2
then G(n, p) is a.a.s Hn(ε, 2)-universal.
Theorem [Conlon, Ferber, N., ˇ Skori´ c ’16] For any constant ∆ 2 N (∆ 3) and ε > 0, if p ✓log3 n n ◆
1 ∆−1
then G(n, p) is a.a.s. Hn(ε, ∆)-universal. Theorem [Conlon, Ferber, N., ˇ Skori´ c ’16] If p ✓log3 n n ◆
1 2−1/2
then G(n, p) is a.a.s Hn(ε, 2)-universal.
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Strategy: embed vertices of H one-by-one by choosing (somehow) a free element from the candidate set
Fact If p ⇣
log n n
1/∆ then G(n, p) a.a.s. has the property that every set of k ∆ vertices has a common neighborhood of size ⇡ npk. Importantly, it is non-empty!! Assume: we can order the vertices of H such that each vertex has ∆ 1 left neighbors (i.e. it is (∆ 1)-degenerate)
Fact If p ⇣
log n n
1/(∆−1) then G(n, p) a.a.s. has the property that every set of k ∆1 vertices has a common neighborhood ⇡ npk. Importantly, it is non-empty!! Assume: we can order the vertices of H such that each vertex has ∆ 1 left neighbors (i.e. it is (∆ 1)-degenerate)
This intuition can be turned into a proof! Theorem For any constants d, ∆ 2 N and ε > 0, if p ✓log2 n n ◆1/d then G(n, p) is a.a.s universal for the family Dn ✓ Hn(ε, ∆) of all d-degenerate graphs.
This intuition can be turned into a proof! Theorem For any constants d, ∆ 2 N and ε > 0, if p ✓log2 n n ◆1/d then G(n, p) is a.a.s universal for the family Dn ✓ Hn(ε, ∆) of all d-degenerate graphs. Remark This is optimal up to the logarithmic factor: consider d-th power of a path on (1 ε)n vertices
This intuition can be turned into a proof! Theorem For any constants d, ∆ 2 N and ε > 0, if p ✓log2 n n ◆1/d then G(n, p) is a.a.s universal for the family Dn(d) ✓ Hn(ε, ∆) of all d-degenerate graphs. The case d = 1 (trees) was considered by Alon, Krivelevich and Sudakov (’07) and independently by Balogh, Csaba, Pei and Samotij (’10)
Preparation: split G ∼ G(n, p) into two parts such that (a) |V1| = (1 − ε/2)n and |V2| = εn/2 (b) G[V1] is Dn(∆ − 1)-universal (c) to be discussed
Preparation: split G ∼ G(n, p) into two parts such that (a) |V1| = (1 − ε/2)n and |V2| = εn/2 (b) G[V1] is Dn(∆ − 1)-universal (c) to be discussed If Hn ∈ Hn(ε, ∆) is (∆ − 1)-degenerate then Hn ⊂ [V1].
Preparation: split G ∼ G(n, p) into two parts such that (a) |V1| = (1 − ε/2)n and |V2| = εn/2 (b) G[V1] is Dn(∆ − 1)-universal (c) to be discussed If Hn ∈ Hn(ε, ∆) is (∆ − 1)-degenerate then Hn ⊂ [V1]. Otherwise: (i) choose a subset S ⊆ V (Hn) such that
(a) Hn − S is (∆ − 1)-degenerate (b) S has a ”nice” structure
Preparation: split G ∼ G(n, p) into two parts such that (a) |V1| = (1 − ε/2)n and |V2| = εn/2 (b) G[V1] is Dn(∆ − 1)-universal (c) to be discussed If Hn ∈ Hn(ε, ∆) is (∆ − 1)-degenerate then Hn ⊂ [V1]. Otherwise: (i) choose a subset S ⊆ V (Hn) such that
(a) Hn − S is (∆ − 1)-degenerate (b) S has a ”nice” structure
(ii) embed Hn − S into G[V1]
Preparation: split G ∼ G(n, p) into two parts such that (a) |V1| = (1 − ε/2)n and |V2| = εn/2 (b) G[V1] is Dn(∆ − 1)-universal (c) to be discussed If Hn ∈ Hn(ε, ∆) is (∆ − 1)-degenerate then Hn ⊂ [V1]. Otherwise: (i) choose a subset S ⊆ V (Hn) such that
(a) Hn − S is (∆ − 1)-degenerate (b) S has a ”nice” structure
(ii) embed Hn − S into G[V1] (iii) somehow embed the vertices from S into V2 (not vertex-by-vertex!!)
Step (i): pick an induced cycle of size at most 2 log n
Step (i): pick an induced cycle of size at most 2 log n
Step (i): pick an induced cycle of size at most 2 log n
Step (ii): embed Hn − S into G[V1]
Step (ii): embed Hn − S into G[V1]
Step (iii): embed S into V2
Step (iii): embed S into V2
Step (iii): embed S into V2
(i) From each connected component which is not (∆ − 1)-degenerate pick an induced cycle of size at most 2 log n and put it into S
(i) From each connected component which is not (∆ − 1)-degenerate pick an induced cycle of size at most 2 log n and put it into S (ii) embed Hn − S into G[V1]
(i) From each connected component which is not (∆ − 1)-degenerate pick an induced cycle of size at most 2 log n and put it into S (ii) embed Hn − S into G[V1] (iii) use Janson’s inequality and Haxell’s hypergraph matching criterion to embed cycles into V2
Theorem [Conlon, Ferber, N., ˇ Skori´ c ’16] For any constant ∆ 2 N (∆ 3) and ε > 0, if p ✓log3 n n ◆
1 ∆−1
then G(n, p) is a.a.s. Hn(ε, ∆)-universal.
Theorem [Conlon, Ferber, N., ˇ Skori´ c ’16] For any constant ∆ 2 N (∆ 3) and ε > 0, if p ✓log3 n n ◆
1 ∆−1
then G(n, p) is a.a.s. Hn(ε, ∆)-universal. Open questions: improve the exponent for ∆ 4 determine the threshold in the degenerate case
Theorem [Conlon, Ferber, N., ˇ Skori´ c ’16] For any constant ∆ 2 N (∆ 3) and ε > 0, if p ✓log3 n n ◆
1 ∆−1
then G(n, p) is a.a.s. Hn(ε, ∆)-universal. Open questions: improve the exponent for ∆ 4 determine the threshold in the degenerate case spanning subgraphs (ε = 0)
G(n, p) has roughly n2p edges G(n, p) is a.a.s Hn(ε, ∆)-universal if p (log3 n/n)1/(∆−1)
G(n, p) has roughly n2p edges G(n, p) is a.a.s Hn(ε, ∆)-universal if p (log3 n/n)1/(∆−1) ) there exists an Hn(ε, ∆)-universal graph G with e(G) = O(n2−1/(∆−1)polylog n)
G(n, p) has roughly n2p edges G(n, p) is a.a.s Hn(ε, ∆)-universal if p (log3 n/n)1/(∆−1) ) there exists an Hn(ε, ∆)-universal graph G with e(G) = O(n2−1/(∆−1)polylog n) Theorem (Alon, Capalbo ’07) There exists an Hn(ε, ∆)-universal graph G with e(G) = O(n2−2/∆)
A graph G is Ramsey for a graph H, G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H
A graph G is Ramsey for a graph H, G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H
A graph G is Ramsey for a graph H, G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H
A graph G is Ramsey for a graph H, G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H
A graph G is Ramsey for a graph H, G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H Theorem (Ramsey ’30) For every graph H there exists N ∈ N such that KN → H.
A graph G is Ramsey for a graph H, G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H Theorem (Ramsey ’30) For every graph H there exists N ∈ N such that KN → H. r(H) = min{N ∈ N : KN → H} ˆ r(H) = min{m ∈ N : ∃G such that e(G) = m and G → H}
r(H) = min{N ∈ N : KN → H} ˆ r(H) = min{m ∈ N : ∃G such that e(G) = m and G → H}
r(H) = min{N ∈ N : KN → H} ˆ r(H) = min{m ∈ N : ∃G such that e(G) = m and G → H} Theorem (Chv´ atal, R¨
edi and Trotter ’83) For every ∆ ∈ N there exists C∆ such that if H is a graph with n vertices and maximum degree ∆ then r(H) ≤ C∆ · n.
r(H) = min{N ∈ N : KN → H} ˆ r(H) = min{m ∈ N : ∃G such that e(G) = m and G → H} Theorem (Chv´ atal, R¨
edi and Trotter ’83) For every ∆ ∈ N there exists C∆ such that if H is a graph with n vertices and maximum degree ∆ then r(H) ≤ C∆ · n. Corollary: ˆ r(H) = O(n2)
r(H) = min{N ∈ N : KN → H} ˆ r(H) = min{m ∈ N : ∃G such that e(G) = m and G → H} Theorem (Kohayakawa, R¨
edi ’11) For every ∆ ∈ N there exists C∆ such that if H is a graph with n vertices and maximum degree ∆ then ˆ r(H) ≤ C∆n2−1/∆ log1/∆ n. Corollary: ˆ r(H) = O(n2)
For every graph H with maximum degree ∆ we have ˆ r(H) ≤ C∆n2−1/∆ log1/∆ n.
For every graph H with maximum degree ∆ we have ˆ r(H) ≤ C∆n2−1/∆ log1/∆ n. R¨
edi ’00: there exists a 3-regular graph H with n vertices and ˆ r(H) ≥ n log1/60 n
For every graph H with maximum degree ∆ we have ˆ r(H) ≤ C∆n2−1/∆ log1/∆ n. R¨
edi ’00: there exists a 3-regular graph H with n vertices and ˆ r(H) ≥ n log1/60 n Theorem (Conlon, N. ’16+) If H is additionally triangle-free then ˆ r(H) ≤ C∆n2−1/(∆−1/2) log1/(∆−1/2) n.
Tn(∆) = family of all triangle-free graphs with n vertices and maximum degree at most ∆ Theorem (Conlon, N. ’16+) – Ramsey-universality If p ✓log n n ◆
1 ∆−1/2
then G ⇠ G(Cn, p) a.a.s has the property that for every red/blue colouring of E(G) = R ∪ B either R or B is Tn(∆)-universal.
Tn(∆) = family of all triangle-free graphs with n vertices and maximum degree at most ∆ Theorem (Conlon, N. ’16+) – Ramsey-universality If p ✓log n n ◆
1 ∆−1/2
then G ⇠ G(Cn, p) a.a.s has the property that for every red/blue colouring of E(G) = R ∪ B either R or B is Tn(∆)-universal. Proof implements previously described strategy in a more difficult setting of sparse regularity.