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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

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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.

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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 at random 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) Erd˝

s-R´

enyi Random Graphs G = {G : G d-regular} Random Regular Graphs G = {G : G tree} Random Trees G = {G : G planar} Random Planar Graphs G = {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

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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

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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

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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

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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-Free Graphs, Graphs that exclude a 2-connected subgraph, Graphs that exclude a cut-point-free graph as a minor, Graphs that admit a Perfect Matching, Graphs with 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

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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-Free Graphs, Graphs that exclude a 2-connected subgraph, Graphs that exclude a cut-point-free graph as a minor, Graphs that admit a Perfect Matching, Graphs with 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

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Connectivity in Bridge-Addable Classes: the conjecture

Intuition: a random graph from a bridge-addable class is likely to be connected:

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 4 / 1

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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 inf

n→∞ Pr (Gn ∈ Gn is connected) ,

should be large.

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 4 / 1

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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 inf

n→∞ 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 (R´

enyi (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

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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, Bollob´ as 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.

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 5 / 1

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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, Bollob´ as 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.

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 5 / 1

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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, Bollob´ as 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

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Connectivity in Bridge-Addable Classes: our results

The McDiarmid-Steger-Welsh Conjecture is true: For every ǫ > 0, there exists an n0 such that for every n ≥ n0 and any bridge-addable class G of graphs, if Gn is non-empty, we have Pr (Gn ∈ Gn is connected) ≥ (1 − ǫ)e−1/2 . (in other words, P(G) ≥ e−1/2) Theorem (Chapuy, P. (2015+)) Furthermore:

∀ǫ > 0 and ∀k ≥ 0, ∃n0 such that for n ≥ n0 one has:

Pr (Gn ∈ Gn has ≤ k + 1 components) ≥ Pr

Poisson

1 2

≤ k
− ǫ.
If P(G) = e−1/2, then Gn locally looks like a forest chosen uniformly at

random among all the ones with n vertices (as n → ∞).

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 6 / 1

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A nice double counting argument for P(G) > e−1

G(i)

= {G ∈ Gn : G has i connected components} We aim to compare the sizes of G(i)

and G(i+1)

: |G(i+1)

| ≤ 1

i |G(i) n |

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1

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A nice double counting argument for P(G) > e−1

G(i)

= {G ∈ Gn : G has i connected components} We aim to compare the sizes of G(i)

and G(i+1)

: |G(i+1)

| ≤ 1

i |G(i) n |

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1

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A nice double counting argument for P(G) > e−1

G(i)

= {G ∈ Gn : G has i connected components} We aim to compare the sizes of G(i)

and G(i+1)

: |G(i+1)

| ≤ 1

i |G(i) n |

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1

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A nice double counting argument for P(G) > e−1

G(i)

= {G ∈ Gn : G has i connected components} We aim to compare the sizes of G(i)

and G(i+1)

: |G(i+1)

| ≤ 1

i |G(i) n |

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1

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A nice double counting argument for P(G) > e−1

G(i)

= {G ∈ Gn : G has i connected components} We aim to compare the sizes of G(i)

and G(i+1)

: |G(i+1)

| ≤ 1

i |G(i) n |

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1

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A nice double counting argument for P(G) > e−1

G(i)

= {G ∈ Gn : G has i connected components} We aim to compare the sizes of G(i)

and G(i+1)

: |G(i+1)

| ≤ 1

i |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−1

i=0 1 i!|G(1) n |

≥ e−1

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1

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Two simplifications

(1) We can assume G is composed by forests (Balister, Bollob´ as 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 implies Pr (Gn ∈ Gn is connected) ≥ e−x . With An = G(1)

and Bn = G(2)

n , it suffices to prove

For every ǫ > 0, there exists an n0 such that for every n ≥ n0 and any bridge-addable class G of forests we have |Bn| ≤ 1 2 + ǫ

· |An| .

Proposition

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 8 / 1

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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 according to 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} BU

n,α = {G ∈ Bn : αG = α, ∃ comp ∼

= U} .

Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 9 / 1

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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 according to 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} BU

n,α = {G ∈ Bn : αG = α, ∃ comp ∼

= U} . (2) Graphs that are adjacent in the bipartite graph, 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

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