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. G n = { G ∈ G : G has n vertices } A random graph from G on n vertices is a graph G n chosen uniformly at random from G n and we denote it as G n ∈ G n . Q : How does G n typically look like? Examples: G = { G : G graph } G ( n , 1 / 2) Erd˝ os-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
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-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
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
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
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 ( G n non-empty for large n ) P ( G ) := lim inf n →∞ Pr ( G n ∈ G n is connected) , should be large. 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 ( G n non-empty for large n ) P ( G ) := lim inf n →∞ Pr ( G n ∈ G n is connected) , should be large. From now on, we will assume that G is a class of labeled graphs. Conjecture (McDiarmid, Steger, Welsh (2006)) For every bridge-addable class G , we have P ( G ) ≥ e − 1 / 2 . - 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
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
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
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
Connectivity in Bridge-Addable Classes: our results Theorem (Chapuy, P. (2015+)) The McDiarmid-Steger-Welsh Conjecture is true: For every ǫ > 0, there exists an n 0 such that for every n ≥ n 0 and any bridge-addable class G of graphs, if G n is non-empty, we have Pr ( G n ∈ G n is connected) ≥ (1 − ǫ ) e − 1 / 2 . (in other words, P ( G ) ≥ e − 1 / 2 ) Furthermore: - ∀ ǫ > 0 and ∀ k ≥ 0, ∃ n 0 such that for n ≥ n 0 one has: � � 1 � � Pr ( G n ∈ G n has ≤ k + 1 components) ≥ Pr Poisson ≤ k − ǫ. 2 - If P ( G ) = e − 1 / 2 , then G n 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
A nice double counting argument for P ( G ) > e − 1 G ( i ) = { G ∈ G n : G has i connected components } n We aim to compare the sizes of G ( i ) and G ( i +1) |G ( i +1) i |G ( i ) | ≤ 1 : n | n n n 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 ) = { G ∈ G n : G has i connected components } n We aim to compare the sizes of G ( i ) and G ( i +1) |G ( i +1) i |G ( i ) | ≤ 1 : n | n n n 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 ) = { G ∈ G n : G has i connected components } n We aim to compare the sizes of G ( i ) and G ( i +1) |G ( i +1) i |G ( i ) | ≤ 1 : n | n n n 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 ) = { G ∈ G n : G has i connected components } n We aim to compare the sizes of G ( i ) and G ( i +1) |G ( i +1) i |G ( i ) | ≤ 1 : n | n n n 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 ) = { G ∈ G n : G has i connected components } n We aim to compare the sizes of G ( i ) and G ( i +1) |G ( i +1) i |G ( i ) | ≤ 1 : n | n n n 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 ) = { G ∈ G n : G has i connected components } n We aim to compare the sizes of G ( i ) and G ( i +1) |G ( i +1) i |G ( i ) | ≤ 1 : n | n n n Pr ( G n ∈ G n is connected) = |G (1) |G (1) |G (1) n | n | n | ≥ e − 1 |G n | = ≥ i =0 |G ( i +1) i ! |G (1) � n − 1 � n − 1 1 | n | n i =0 Guillem Perarnau Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture 7 / 1
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