jephian c.-h. lin april 21, 2016 presentation for math...

On the tree-depth of random graphs

Jephian C.-H. Lin

Department of Mathematics, Iowa State University

April 21, 2016Presentation for Math 608

On the tree-depth of random graphs 1/27 Department of Mathematics, Iowa State University

Presented paper

G. Perarnau and O. SerraOn the tree-depth of random graphs. Discrete AppliedMathematics. 168(2014) 119–126.


Main results

Theorem 1.Let G ∼ G(n,p) be a random graph with np →∞, then whp

td(G) = n −O (√

n

p) .

Theorem 2.Let G ∼ G(n,p) be a random graph with p = c

n , with c > 0.

1. If c < 1, then whp td(G) = Θ(log log n).2. If c = 1, then whp td(G) = Θ(log n).3. If c > 1, then whp td(G) = Θ(n).


Tree-depth: tree closure

Let T be a rooted tree. The height of T is the number ofvertices of the longest rooted path.

The closure of T is the graph obtained from T by addingedges to all ancestor-descendant pair.


Tree-depth

Let G be a connected graph. Then T is an elimination tree ofG if G is a subgraph of the closure of T .

The tree-depth of G , denoted as td(G), is the minimumheight of an elimination tree of G .

For complete graphs, td(Kn) = n.

For paths, td(Pn) = ⌊log2 n⌋ + 1.


Upper bound of tree-width

Lemma 3.For any tree T ,

td(T ) ≤ ⌊log2 n⌋ + 1.

Corollary 4.

Let G be a graph. If G − S is a tree, then

td(G) ≤ ⌊log2 n⌋ + 1 + ∣S ∣.


Lower bound for tree-depth

Lemma 5.Let G be a graph with diameter d . Then

td(G) ≥ log d .

Proof.The graph G contains Pd+1 as a subgraph, so

td(G) ≥ td(Pd+1) ≥ ⌊log(d + 1)⌋ + 1 ≥ log d ,

since tree-depth is minor monotone.


Tree-width: k-tree

A k-tree is a graph starting with Kk+1 and inductively addingone vertex at a time, joining with an existing k-clique.

The tree-width of a graph G is the minimum k such that G isa subgraph of a k-tree.

k = 3


tree-depth and tree-width

Proposition 6.

For all graph G ,tw(G) ≤ td(G).

Proof.Let T be an elimination tree with height k . Then the closure of Tis a subgraph of a k-tree.


Balance k-partition

Let G be a graph on n vertices. A balanced k-partition of Gis a partition of V (G) = A∪S ∪B such that ∣S ∣ = k + 1 and noedges between A and B and

1

3(∣V ∣ − ∣S ∣) ≤ ∣A∣, ∣B ∣ ≤ 2

3(∣V ∣ − ∣S ∣).

S∣S ∣=k+1A B


Lower bound of tree-width

Lemma 7.Let G be a graph on n vertices. Then for any k withtw(G) ≤ k ≤ n − 4, G has a balanced k-partition.

Corollary 8.

If k ≤ n − 4 and G contains no balanced k-partition, thentw(G) ≥ k + 1.



td(G) = n −O (√

n

p) .


Proof of Theorem 1

Assume p = c(n)/n with c(n)→∞.

Let f (c) = 3√

ln 3c . (Will see the reason later.)

Let k + 1 = n − f (c)n.

Claim: that whp there is no balanced k-partition.

If the Claim is true, then

td(G) ≥ tw(G) ≥ n − f (c)n = n −O(√

n

p) = n − o(n).


Proof of the Claim

Suppose (A,S ,B) is a possible balanced k-partition. Then∣S ∣ = k + 1, ∣A∣ + ∣B ∣ = n − k − 1 = f (c)n, and

f (c)n3

≤ ∣A∣, ∣B ∣ ≤ 2f (c)n3

.

Then ∣A∣∣B ∣ ≥ 2f (c)2

9 n2.

Let X(A,S ,B) be the event that (A,S ,B) is a balancedk-partition. Then


Pr( ⋃A,S ,B

partition V (G)

X(A,S ,B)) ≤ ∑A,S ,B

partition V (G)

Pr(X(A,S ,B))

(as ∣A∣∣B ∣ ≥ 2f (c)29

n2) ≤ 3n(1 − p)2f (c)2

9n2

(as 1 − x ≤ e−x) ≤ exp(ln 3)n − 2f (c)29

n2p .

(f (c) = 3

√ln 3

c) = exp−(ln 3)n.

This probability goes to zero as n → 0.



n , with c > 0.



Case 1 Upper Bound

By Corollary 4, if G is a tree or unicyclic graph on n vertices,then td(G) ≤ log n + 2.

By [Erdos and Renyi 1960], when 0 < c < 1, whp G iscomposed of trees and unicyclic graphs, with the order of thelargest component Θ(log n).

Thus td(G) = O(log log n).


Case 1 Lower Bound

In [Erdos and Renyi 1960], the number of trees of order k inG(n,p) follows a normal distribution with µ =Mk andσ =

√Mk .

When k = log n, Mk →∞, so whp there are many tree oforder log n.

By [Renyi and Szekeres 1967], a random tree on k verticeshas diameter Θ(

√k).

In G(n,p), consider all trees with size log n. Then whp thereis a tree T with diameter d = Θ(

√log n).

td(G) ≥ td(T ) ≥ Ω(log d) = Ω(log(√

log n)) = Ω(log log n).


Case 2 Upper Bound

A (k , `)-component is a connected graph with k vertices andk + ` edges.

If G is a (k , `)-component, then td(G) ≤ log n + `. By [Erdos and Renyi 1960], when c = 1, whp the order of the

largest component Θ(n 23 ).

It can be shown whp every component is a (k , `)-component

with k = O(n 23 ) and ` = O(log log n).

Thus td(G) = log k + ` ≤ O(log n) +O(log log n) = O(log n).


Case 2 Lower Bound

Theorem 9 (Nachmias and Peres 2008).

Let C be the largest component of a random graph in G(n, 1n).Then for any ε > 0, there exists A = A(ε) such that

Pr(diam(C) ∉ [A−1n13 ,An

13 ]) < ε.

The Theorem shows Pr(diam(C) < n13−ε) = o(1). This means whp

td(G) = Ω(log n13−ε) = Ω(log n).


Case 3 Upper Bound

For any graph td(G) ≤ n, so td(G) = O(n).


Case 3 History

Let G ∼ G(n,p) with p = cn .

Kloks 1994 shows tw(G) = Θ(n) for c > 2.36.

Gao 2006 shows improves to c > 2.162 and conjectured forsome threshold 1 < c < 2 tree-width is linear.

Lee, Lee, and Oum 2012 proved the conjecture.


Case 3 Lower Bound: Another approach

The edgewise Cheeger constant of a graph G is a

Φ(G) = minX⊆V

0<∣X ∣≤ n2

e(X ,V −X )d(X ) , where d(X ) = ∑

x∈X

d(x).

Let G ∼ G(n, cn) with c > 1. Pick α, δ > 0. [Benjaminiet. al. 2006] shows whp there is a subgraph H with Φ(H) ≥ αand ∣V (H)∣ ≥ δn.

Will show that whp

td(G) ≥ tw(G) ≥ tw(H) ≥ γ0n,

for some γ0.


Let tw(H) = k . Pick a balanced k-partition (A,S ,B). Then

α ≤ e(A,V (H) −A)d(A) ≤ e(A,S)

∣A∣ .

d(S) ≥ e(A,S) ≥ α∣A∣ ≥ α(δn − ∣S ∣)3

.

It is enough to show that with ∣S ∣ ≤ γ0n, whp there is no suchset S .

Let ∣S ∣ = γn. Want to find d(S) ≥ α(δ−γ)3 n =∶ βn.

Pr(∃S ∶ ∣S ∣ = γn,d(S) ≥ βn) ≤ ( n

γn)γn2

∑e=βn

(γn2

e)pe(1 − p)γn2−e

≤ γn2 (( eγ)γ

(γecβ

)β

)n

= γn2f (γ)n.

Pick γ0 small enough such that f (γ) < 1. Then the proof iscompleted.



td(G) = n −O (√

n

p) .


n , with c > 0.


Thank you!


References I

I.Benjamini, G.Kozma, and N.Wormald. The mixing time ofthe giant component of a random graph, 2006.

P.Erdos and A.Renyi. On the evolution of random graphs.Magyar Tud. Akad. mat. Kutato Int. Kozl. 5(1960) 17–61.

Y.Gao. On the threshold of having a linear treewidth in randomgraphs. In: Computing and Combinatorics. In: Lecture Notesin Comput. Sci. Vol 4112, Springer, Berlin, 2006, pp. 226–234.

T.Kloks. Treewidth: Computations and Approximations. In:Lecture Notes in Computer Science. Vol 842, Springer,-Verlag,Berlin, 1994.

C.Lee, J.Lee, and S.i. Oum. Rank-width of random graphs.J. Graph Theory 70(2012) 339–347.


References II

A.Nachmias and Y.Peres. Critical random graphs: diameterand mixing time. Ann. Probab. 36(2008) 1267–1286.

A.Renyi and G.Szekeres. On the height of trees.J. Aust. Math. Soc. 7(1967) 497–507.


jephian c.-h. lin april 21, 2016 presentation for math...

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