connectivity in bridge-addable graph classes: the ...rsa 2015,pittsburgh, pa- 31st july, 2015 mcgill...

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:

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.

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.

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.

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.

P(G) ≥ e−1/2 .

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.

P(G) ≥ e−1/2 .

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.

Norin (2013):

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

)≤ 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→∞).

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

n | ≤ 1i|G(i)n |

=|G(1)n |∑n−1

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

≥ |G(1)n |∑n−1i=0

1i!|G(1)n |

≥ e−1

n | ≤ 1i|G(i)n |

=|G(1)n |∑n−1

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

≥ |G(1)n |∑n−1i=0

1i!|G(1)n |

≥ e−1

n | ≤ 1i|G(i)n |

=|G(1)n |∑n−1

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

≥ |G(1)n |∑n−1i=0

1i!|G(1)n |

≥ e−1

n | ≤ 1i|G(i)n |

=|G(1)n |∑n−1

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

≥ |G(1)n |∑n−1i=0

1i!|G(1)n |

≥ e−1

n | ≤ 1i|G(i)n |

=|G(1)n |∑n−1

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

≥ |G(1)n |∑n−1i=0

1i!|G(1)n |

≥ e−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| ≤(

)· |An| .

Proposition

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} .

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|

Sketch of the proof

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)

Open Problems

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)

Open Problems

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

connectivity in bridge-addable graph classes: the ...rsa 2015,pittsburgh, pa- 31st july, 2015 mcgill...

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