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MATH 7018: Homework 2 (due Feb 13 at 11:59pm)
Please typeset your homework solutions using LATEX.
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Problem 1.
Let G be a graph with n ě 2 vertices and m edges. Recall that a cut in G is a pair C “ tA, Bu of disjoint nonempty subsets of V pGq such that A Y B “ V pGq. The set of all edges joining A to B is denoted by EpCq. Let maxcutpGq :“ maxC |EpCq|, where C ranges over all cuts in G. ( a ) Use a simple probabilistic argument to show that maxcutpGq ě m{ 2.
In the remainder of this problem our goal is to improve the bound in ( a ) by an amount of order Θp
mq. Let k :“ χpGq be the chromatic number of G and fix a partition V pGq “ V 1 ... \ V k of the vertex set of G into k independent sets. Define t :“ tk{ 2 u and form a subset A Ď V pGq by picking a uniformly random t-element set I Ă rks and setting A :“
i P I V i. ( b ) Use the above construction to show that
maxcutpGq ě
`
2 k
m.
( c ) Deduce from part ( b ) that
maxcutpGq ě
m 2
`
8 m ` 1 ´ 1 8
Problem 2.
The goal of this problem is to prove the following:
Theorem. Let G be an n-vertex graph with average degree d, where n ě 1. Suppose that positive integers q, r, s, t P N`^ satisfy the inequality d t n t ´^1
n r
q n
¯ t ě s.
Then there exists a set S Ď V pGq of size |S| ě s such that every r vertices from S have at least q ` 1 common neighbors in G.
Let G, n, d, q, r, s, t be as in the theorem. For a set A Ď V pGq, let N JAK denote the set of all common neighbors of the vertices in A. Pick x 1 ,... , x t P V pGq independently and uniformly at random and let U :“ N Jx 1 ,... , x t K. Define random variables X :“ |U | and Y :“ |tR Ď U : |R| “ r and |N JRK| ď qu|.
( a ) Show that ErXs ě
d t n t ´^1
and ErY s ď
n r
q n
¯ t .
( b ) Complete the proof of the theorem.
Remark. This proof technique is known as dependent random choice. It has been applied to a large number of problems in extremal combinatorics.
Bonus (not for credit). Here is an application of the above theorem in extremal graph theory. The extremal number expn, Hq (also known as the Turán number ) of a finite graph H is the maximum number of edges in an n-vertex graph G that has no subgraph (not necessarily induced) isomorphic to H. Prove the following:
Corollary. If H is a bipartite graph with a bipartition pA, Bq such that every vertex in A has degree at most t, then expn, Hq “ Opn^2 ´^1 { t q.
This is a generalization of the Kővári–Sós–Turán theorem (which considers the special case when H is the complete bipartite graph K s,t .)
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Problem 3.
The Chernoff bound says that if X 1 ,... , X n are mutually independent Bernoulli random variables and X :“
ř n i “ 1 X i , then for all^ λ^ ą^0 , P
|X ´ ErXs| ě λ
n
ď 2 e´ λ
(^2) { 2 . On the other hand, even if we only assume that X 1 ,... , X n are pairwise independent, then P
|X ´ ErXs| ě λ
n
ď Opλ´^2 q (˚) by Chebyshev’s inequality. In this problem you will show that the dependence on λ in (˚) cannot be improved in general (hence Chernoff fails if we only assume pairwise independence). Let Z 1 ,... , Z k be mutually independent random variables equal to 1 or ´ 1 with probability 1 { 2. Set n :“ 2 k^ ´ 1 and list all nonempty subsets of t 1 ,... , ku as S 1 ,... , S n. Define X i as follows:
X i :“
1 `
ź
j P Si
Z j
( a ) Show that X 1 ,... , X n are pairwise independent Bernoulli random variables.
( b ) Setting X :“
ř n i “ 1 X i^ and^ λ^ “
n{ 2 , show that P
|X ´ ErXs| ě λ
n
“ Θpλ´^2 q.
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Problem 4.
Suppose we throw m balls into n bins independently and uniformly and random. Let X be the random variable equal to the number of bins that remain empty.
( a ) Show that for any constant ε P p 0 , 1 q,
n^ limÑ8 ErXs “
8 if m ď p 1 ´ εqn log n, 0 if m ě p 1 ` εqn log n.
( b ) Show that for any constant ε P p 0 , 1 q,
n^ limÑ8 Prat least one bin is emptys “
1 if m ď p 1 ´ εqn log n, 0 if m ě p 1 ` εqn log n.
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