Exercise 1. An Eulerian circuit of G is a sequence of vertices v1 v2 . . . v` (a vertex may
appear more than once) so that every edge uw 2 E(G) appears as vi vi+1 for some i
in the sequence, and so that v1 = v` . A Eulerian graph is one which has a Eulerian
circuit.
A Hamiltonian cycle of graph G on atleast three vertices is an sequence v1 v2 . . . vn
such that each u 2 V (G) appears exactly once, v1 = v2 and each vi vi+1 2 E(G). A
graph is Hamiltonian if it has a Hamiltonian cycle.
(a) Let A be the set of Eulerian graphs. Show that A is not monotone.
(b) Let B be the set of Hamiltonian graphs. Is B monotone?
Exercise 2. A graph G with n
3 vertices, denoted Cn , is a cycle if its vertices can be
(re)-labelled v1 , . . . , vn such that E(G) = {vi vi+1 : i 2 [n]} where the subscript addition is taken modulo n. For example a cycle on 3 vertices is
and there are three
cycles on four vertices , , .
A connected graph is one in which any two vertices uv are connected by a sequence
of vertices v1 . . . v` so that u = v1 , v = v` and each vi vi+1 is an edge. For example
is connected but
is not connected.
(a) A graph with n vertices and n edges must contain a cycle as a subgraph.
(b) A connected graph with n vertices and n edges must contain exactly one cycle.
(c) Give an example to show that the assumption of connectivity is needed for part b.
Exercise 3. (Covered in lectures) Let A4 be the set of all graphs which contain
subgraph.
as a
(a) Show that P(G(n, 1/2) 2 A4 ) ! 1.
(b) (optional) Fix a constant 0 < p  1, and show that P(G(n, p) 2 A4 ) ! 1.
Exercise 4. (Covered in lectures) Prove the following:
Let X1 , X2 , . . . be a sequence of random variables each taking non-negative values.
If E[Xn ] ! 0 then
P(Xn = 0) ! 1,
and if E[Xn ] > 0 for each n, and V[Xn ]/E[Xn ] ! 0 then
P(Xn = 0) ! 0.
Exercise 5. Show whp np ! 1 implies whp Gn contains
i.e. a 3-cycle1 .
Let Yn count the number of
in Gn and for any 3-subset of vertices S ⇢ V (G) let
AS be the event that Gn restricted to the vertices S is a .
(a) Show by linearity of expectation that:
X ✓
V(Yn ) =
P(AS & 1AT )
S,T 2([n]
3 )
◆
P(AS )P(AT ) .
1 This exercise demonstrates a di↵erent way to prove the second part of Theorem ??. In the proof we showed
that whp e(Gn ) n for np ! 1 and from this and Q 2a we concluded that np ! 1 implies whp Gn has a cycle.
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(b) Notice that when the sets of vertices S and T don’t intersect that the events AS and AT
are independent. What about when they intersect on one vertex? Using (a) show that:
X
V(Yn ) 
P(AS & AT ).
|S\T |={2,3}
(c) After some case analysis and from (b) show: V(Yn )  n4 p5 + n3 p3 .
(d) From (c) conclude that whp Yn > 0. Hint: use Chebyshev’s inequality.
Exercise 6. Given k 2 N, let Pk be the set of graphs which have a path on k vertices as a
subgraph.
(a) (Covered in lectures) Find the threshold function for P3 (notice P3 is the set of graphs
containing the path
as a subgraph).
(b) Find the threshold for P4 .
(c) (optional) Let k 2 N be a constant. Find the threshold for Pk in terms of k and n.
Exercise 7. For each of the following boolean functions f , aka voting schemes, find a set
S such that the function is expressible in terms of that character, i.e. f (x) = S (x)
or f (x) =
S (x).
(a) The dictator function, Dict1n (x) = x1 .
(b) The parity function, P ar(x).
(c) The XOR function of the first two inputs, f (x) = XOR(x1 , x2 ).
(d) The constant function f (x) = 1.
Exercise 8. We can define an interated majority function for n = 3k . The base case is
Imaj1 (x1 , x2 , x3 ) = Maj3 (x1 , x2 , x3 ) and
Imajk (x) = Maj3 (Imajk
1 (x1 , . . . , x3k
1
), Imajk
1 (x3k
1 +1
, . . . , x2.3k 1 ), IMajk
1 (x2.3k
1 +1
, . . . , x3k )).
For example, for k = 2, Imaj2 (x1 , . . . , x9 ) = Maj3 (Maj3 (x1 , x2 , x3 ), Maj3 (x4 , x5 , x6 ), Maj3 (x7 , x8 , x9 )).
(a) Calculate the influence of the i-th bit Ii (Imaj2 ) and total influence I p (Imaj2 ).
(b) Can you calculate Iip (Imajk ) and I p (Imajk )? You may take p = 1/2 if you like.
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