connectivity in bridge-addable graph classes: the ...rsa 2015,pittsburgh, pa- 31st july, 2015 mcgill...
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Connectivity in bridge-addable graph classes:the McDiarmid-Steger-Welsh conjecture
Guillem Perarnau
RSA 2015, Pittsburgh, PA - 31st July, 2015
McGill University, Montreal, Canada
joint work with Guillaume Chapuy.
Random graphs in a class
Let G be a class of graphs.
Gn = {G ∈ G : G has n vertices}
A random graph from G on n vertices is a graph Gn chosen uniformly atrandom from Gn and we denote it as Gn ∈ Gn.
Q: How does Gn typically look like?
Examples:
G = {G : G graph} G(n, 1/2) Erdos-Renyi Random GraphsG = {G : G d-regular} Random Regular GraphsG = {G : G tree} Random TreesG = {G : G planar} Random Planar GraphsG = {G : G triangle-free} Random Triangle-Free Graphs
......
Study random graphs from a class G that satisfies some mild condition
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 2 / 1
Bridge-Addable Classes
A class G of graphs is bridge-addable if the following is true:
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 3 / 1
Bridge-Addable Classes
A class G of graphs is bridge-addable if the following is true:
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 3 / 1
Bridge-Addable Classes
A class G of graphs is bridge-addable if the following is true:
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 3 / 1
Bridge-Addable Classes
A class G of graphs is bridge-addable if the following is true:
Examples: Forests, Planar Graphs, Graphs with bounded genus, Triangle-FreeGraphs, Graphs that exclude a 2-connected subgraph, Graphs that exclude acut-point-free graph as a minor, Graphs that admit a Perfect Matching, Graphswith bounded Treewidth, All Graphs, Connected Graphs.
Non-Examples: Regular graphs, Graphs with m edges, Non-connected Graphs.
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 3 / 1
Bridge-Addable Classes
A class G of graphs is bridge-addable if the following is true:
Examples: Forests, Planar Graphs, Graphs with bounded genus, Triangle-FreeGraphs, Graphs that exclude a 2-connected subgraph, Graphs that exclude acut-point-free graph as a minor, Graphs that admit a Perfect Matching, Graphswith bounded Treewidth, All Graphs, Connected Graphs.
Non-Examples: Regular graphs, Graphs with m edges, Non-connected Graphs.
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 3 / 1
Connectivity in Bridge-Addable Classes: the conjecture
Intuition: a random graph from a bridge-addable class is likely to be connected:
From now on, we will assume that G is a class of labeled graphs.
For every bridge-addable class G, we have
P(G) ≥ e−1/2 .
Conjecture (McDiarmid, Steger, Welsh (2006))
- If F is the class of all forests, then P(F) = e−1/2 (Renyi (1959)).
- If C is the class of all connected graphs, then P(C) = 1.
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 4 / 1
Connectivity in Bridge-Addable Classes: the conjecture
Intuition: a random graph from a bridge-addable class is likely to be connected:
In other words, if G is bridge-addable (Gn non-empty for large n)
P(G) := lim infn→∞
Pr (Gn ∈ Gn is connected) ,
should be large.
From now on, we will assume that G is a class of labeled graphs.
For every bridge-addable class G, we have
P(G) ≥ e−1/2 .
Conjecture (McDiarmid, Steger, Welsh (2006))
- If F is the class of all forests, then P(F) = e−1/2 (Renyi (1959)).
- If C is the class of all connected graphs, then P(C) = 1.
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 4 / 1
Connectivity in Bridge-Addable Classes: the conjecture
Intuition: a random graph from a bridge-addable class is likely to be connected:
In other words, if G is bridge-addable (Gn non-empty for large n)
P(G) := lim infn→∞
Pr (Gn ∈ Gn is connected) ,
should be large.
From now on, we will assume that G is a class of labeled graphs.
For every bridge-addable class G, we have
P(G) ≥ e−1/2 .
Conjecture (McDiarmid, Steger, Welsh (2006))
- If F is the class of all forests, then P(F) = e−1/2 (Renyi (1959)).
- If C is the class of all connected graphs, then P(C) = 1.
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 4 / 1
Connectivity in Bridge-Addable Classes: previous results
Results on the conjecture:
McDiarmid, Steger and Welsh (2006):
For every bridge-addable class G, we have P(G) ≥ e−1.
Balister, Bollobas and Gerke (2008):
For every bridge-addable class G, we have P(G) ≥ e−0.7983.
Norin (2013):
For every bridge-addable class G, we have P(G) ≥ e−2/3.
The conjecture on more restricted graph classes:
A class has girth at least k if all the graphs in it have girth at least k.
Addario-Berry and Reed (2007):
For every bridge-addable class G with large girth, we have P(G) ≥ e−1/2.
A class is bridge-alterable if it is stable under bridge addition and deletion.
Addario-Berry, McDiarmid and Reed (2012), Kang and Panagiotou (2013):
For every bridge-alterable class G, we have P(G) ≥ e−1/2.
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 5 / 1
Connectivity in Bridge-Addable Classes: previous results
Results on the conjecture:
McDiarmid, Steger and Welsh (2006):
For every bridge-addable class G, we have P(G) ≥ e−1.
Balister, Bollobas and Gerke (2008):
For every bridge-addable class G, we have P(G) ≥ e−0.7983.
Norin (2013):
For every bridge-addable class G, we have P(G) ≥ e−2/3.
The conjecture on more restricted graph classes:
A class has girth at least k if all the graphs in it have girth at least k.
Addario-Berry and Reed (2007):
For every bridge-addable class G with large girth, we have P(G) ≥ e−1/2.
A class is bridge-alterable if it is stable under bridge addition and deletion.
Addario-Berry, McDiarmid and Reed (2012), Kang and Panagiotou (2013):
For every bridge-alterable class G, we have P(G) ≥ e−1/2.
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 5 / 1
Connectivity in Bridge-Addable Classes: previous results
Results on the conjecture:
McDiarmid, Steger and Welsh (2006):
For every bridge-addable class G, we have P(G) ≥ e−1.
Balister, Bollobas and Gerke (2008):
For every bridge-addable class G, we have P(G) ≥ e−0.7983.
Norin (2013):
For every bridge-addable class G, we have P(G) ≥ e−2/3.
The conjecture on more restricted graph classes:
A class has girth at least k if all the graphs in it have girth at least k.
Addario-Berry and Reed (2007):
For every bridge-addable class G with large girth, we have P(G) ≥ e−1/2.
A class is bridge-alterable if it is stable under bridge addition and deletion.
Addario-Berry, McDiarmid and Reed (2012), Kang and Panagiotou (2013):
For every bridge-alterable class G, we have P(G) ≥ e−1/2.
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 5 / 1
Connectivity in Bridge-Addable Classes: our results
The McDiarmid-Steger-Welsh Conjecture is true:
For every ε > 0, there exists an n0 such that for every n ≥ n0 and anybridge-addable class G of graphs, if Gn is non-empty, we have
Pr (Gn ∈ Gn is connected) ≥ (1− ε)e−1/2 .
(in other words, P(G) ≥ e−1/2)
Theorem (Chapuy, P. (2015+))
Furthermore:
- ∀ε > 0 and ∀k ≥ 0, ∃n0 such that for n ≥ n0 one has:
Pr (Gn ∈ Gn has ≤ k + 1 components) ≥ Pr
(Poisson
(1
2
)≤ k
)− ε.
- If P(G) = e−1/2, then Gn locally looks like a forest chosen uniformly atrandom among all the ones with n vertices (as n→∞).
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 6 / 1
A nice double counting argument for P(G) > e−1
G(i)n = {G ∈ Gn : G has i connected components}
We aim to compare the sizes of G(i)n and G(i+1)n : |G(i+1)
n | ≤ 1i|G(i)n |
Pr (Gn ∈ Gn is connected) =|G(1)n ||Gn|
=|G(1)n |∑n−1
i=0 |G(i+1)n |
≥ |G(1)n |∑n−1i=0
1i!|G(1)n |
≥ e−1
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1
A nice double counting argument for P(G) > e−1
G(i)n = {G ∈ Gn : G has i connected components}
We aim to compare the sizes of G(i)n and G(i+1)n : |G(i+1)
n | ≤ 1i|G(i)n |
Pr (Gn ∈ Gn is connected) =|G(1)n ||Gn|
=|G(1)n |∑n−1
i=0 |G(i+1)n |
≥ |G(1)n |∑n−1i=0
1i!|G(1)n |
≥ e−1
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1
A nice double counting argument for P(G) > e−1
G(i)n = {G ∈ Gn : G has i connected components}
We aim to compare the sizes of G(i)n and G(i+1)n : |G(i+1)
n | ≤ 1i|G(i)n |
Pr (Gn ∈ Gn is connected) =|G(1)n ||Gn|
=|G(1)n |∑n−1
i=0 |G(i+1)n |
≥ |G(1)n |∑n−1i=0
1i!|G(1)n |
≥ e−1
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1
A nice double counting argument for P(G) > e−1
G(i)n = {G ∈ Gn : G has i connected components}
We aim to compare the sizes of G(i)n and G(i+1)n : |G(i+1)
n | ≤ 1i|G(i)n |
Pr (Gn ∈ Gn is connected) =|G(1)n ||Gn|
=|G(1)n |∑n−1
i=0 |G(i+1)n |
≥ |G(1)n |∑n−1i=0
1i!|G(1)n |
≥ e−1
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1
A nice double counting argument for P(G) > e−1
G(i)n = {G ∈ Gn : G has i connected components}
We aim to compare the sizes of G(i)n and G(i+1)n : |G(i+1)
n | ≤ 1i|G(i)n |
Pr (Gn ∈ Gn is connected) =|G(1)n ||Gn|
=|G(1)n |∑n−1
i=0 |G(i+1)n |
≥ |G(1)n |∑n−1i=0
1i!|G(1)n |
≥ e−1
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1
A nice double counting argument for P(G) > e−1
G(i)n = {G ∈ Gn : G has i connected components}
We aim to compare the sizes of G(i)n and G(i+1)n : |G(i+1)
n | ≤ 1i|G(i)n |
Pr (Gn ∈ Gn is connected) =|G(1)n ||Gn|
=|G(1)n |∑n−1
i=0 |G(i+1)n |
≥ |G(1)n |∑n−1i=0
1i!|G(1)n |
≥ e−1
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1
Two simplifications
(1) We can assume G is composed by forests (Balister, Bollobas and Gerke).
(2) If there exists x ≥ 0 such that |G(2)n | ≤ x · |G(1)n |, then |G(i+1)n | ≤ x
i· |G(i)n |,
which impliesPr (Gn ∈ Gn is connected) ≥ e−x .
With An = G(1)n and Bn = G(2)n , it suffices to prove
For every ε > 0, there exists an n0 such that for every n ≥ n0 and anybridge-addable class G of forests we have
|Bn| ≤(
1
2+ ε
)· |An| .
Proposition
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 8 / 1
A local double-counting approach
Our approach is local in two senses.Fix T0 and U0, sets of rooted and unrooted trees.
(1) We partition the sets An and Bn accordingto local statistics:
∀T ∈ T0
αG (T ) = # pendant copies of T in G .
For α ∈ ZT0 and U ∈ U0, we define
An,α = {G ∈ An : αG = α}
BUn,α = {G ∈ Bn : αG = α,∃ comp ∼= U} .
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 9 / 1
A local double-counting approach
Our approach is local in two senses.Fix T0 and U0, sets of rooted and unrooted trees.
(1) We partition the sets An and Bn accordingto local statistics:
∀T ∈ T0
αG (T ) = # pendant copies of T in G .
For α ∈ ZT0 and U ∈ U0, we define
An,α = {G ∈ An : αG = α}
BUn,α = {G ∈ Bn : αG = α,∃ comp ∼= U} .
(2) Graphs that are adjacent in the bipartitegraph, have similar statistics:
If (G ,G ′) is an edge, then
‖αG − αG ′‖ ≤ max
U∈U0|U|
.
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 9 / 1
Sketch of the proof
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 10 / 1
Sketch of the proof
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 10 / 1
Sketch of the proof
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 10 / 1
Sketch of the proof
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 10 / 1
Sketch of the proof
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 10 / 1
Sketch of the proof
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 10 / 1
Sketch of the proof
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 10 / 1
Open Problems
Let F be the class of all forests. Non-asymptotic version of MSW Conjecture:
For every n ≥ 1 and any bridge-addable class G we have,
Pr (Gn ∈ Gn is connected) ≥ Pr (Fn ∈ Fn is connected) .
Conjecture (Addario-Berry, McDiarmid and Reed)
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 11 / 1
Open Problems
Let F be the class of all forests. Non-asymptotic version of MSW Conjecture:
For every n ≥ 1 and any bridge-addable class G we have,
Pr (Gn ∈ Gn is connected) ≥ Pr (Fn ∈ Fn is connected) .
Conjecture (Addario-Berry, McDiarmid and Reed)
All our results are for classes of labeled graphs.
Does there exist a δ > 0 and n0 such that for every n ≥ n0 and everybridge-addable class of unlabeled graphs G we have
Pr (Gn ∈ Gn is connected) > δ ?
Open problem (McDiarmid)
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 11 / 1
Open Problems
Let F be the class of all forests. Non-asymptotic version of MSW Conjecture:
For every n ≥ 1 and any bridge-addable class G we have,
Pr (Gn ∈ Gn is connected) ≥ Pr (Fn ∈ Fn is connected) .
Conjecture (Addario-Berry, McDiarmid and Reed)
All our results are for classes of labeled graphs.
Does there exist a δ > 0 and n0 such that for every n ≥ n0 and everybridge-addable class of unlabeled graphs G we have
Pr (Gn ∈ Gn is connected) > δ ?
Open problem (McDiarmid)
THANKS FOR YOUR ATTENTION
Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 11 / 1
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