MATH 3012 H, Test 2, Answer sheet, Page 1 of 4
Problem 1. All parameters in the questions below are assumed to be integers. Write down a
closed-form expression (which can be just a number) for
• the maximum number of edges that a planar graph with
n vertices can have (n > 3).
3n − 6
• the largest chromatic number that a planar graph can have.
4
• the maximum number of edges that a triangle-free graph with
n vertices can have (n > 2 and n is even).
n2
4
• the smallest integer d such that every graph with all vertices of
degree at least d has a Hamilton cycle (n > 3).
n
2
• the smallest clique number that a graph with chromatic number
k can have (k > 3).
2
• the number of labeled trees on n vertices with labels 1, 2, . . . , n
(n > 2).
nn−2
• the value φ(p3 ) of the Euler function for p3 , where p is a prime
number (recall: φ(n) = |{k ∈ N : 1 6 k 6 n and gcd(n, k) = 1}|).
p3 − p2
• the width of the subset lattice (2X , ⊆), where X = {1, 2, . . . , n}
(n > 1).
n
n
2
• the number of maximal (longest) chains in the subset lattice
(2X , ⊆), where X = {1, 2, . . . , n} (n > 1).
n!
• (Bonus) the maximum number of elements that a poset of
width w and height h can have (w, h > 1).
wh
Problem 2. Let X be a finite set and A1 , A2 , A3 ⊆ X. Write down the inclusion-exclusion formula
• for |X − (A1 ∪ A2 ∪ A3 )| using |X| and the sizes of appropriate intersections of A1 , A2 , A3 .
|X| − |A1 | − |A2 | − |A3 | + |A1 ∩ A2 | + |A1 ∩ A3 | + |A2 ∩ A3 | − |A1 ∩ A2 ∩ A3 |
• for the number of surjective (onto) functions f : {1, 2, . . . , n} → {1, 2, 3}, where n > 3.
3n − 3 · 2n + 3
MATH 3012 H, Test 2, Answer sheet, Page 2 of 4
Problem 3. Consider the poset P on six elements a, b, c, d, e, f drawn in figure A below.
c
c
f
e
b
a
e
b
a
d
c
f
e
b
a
d
A
f
d
B
C
• Draw the comparability graph of P using the vertices in figure B above.
• Draw the incomparability graph of P using the vertices in figure C above.
• (1.5 points) Write down the height h of P , an example of a chain of size h in P , and an example
of a partition of P into h antichains.
h = 3,
chain: {a, b, c},
{a, d}, {b, e}, {c, f }
antichain partition:
• (1.5 points) Write down the width w of P , an example of an antichain of size w in P , and an
example of a partition of P into w chains.
w = 2,
antichain: {a, d},
chain partition:
{a, e, f }, {d, b, c}
• (2 points) The poset P is an interval order. Draw the minimal interval representation of P
(that is, the representation using the minimum number of distinct coordinates) in the space
below. Mark clearly the coordinates of the endpoints of the intervals.
D1 = D(a) = D(d) = ∅
U1 = U (a) = {b, c, e, f }
D2 = D(e) = {a}
U2 = U (d) = {b, c, f }
D3 = D(b) = {a, d}
U3 = U (e) = {c, f }
D4 = D(f ) = {a, d, e}
U4 = U (b) = {c}
D5 = D(c) = {a, b, d, e}
U5 = U (c) = U (f ) = ∅
d
I(b) = [3, 4]
I(c) = [5, 5]
I(d) = [1, 2]
I(e) = [2, 3]
I(f ) = [4, 5]
b
a
2
c
f
e
1
I(a) = [1, 1]
3
4
5
MATH 3012 H, Test 2, Answer sheet, Page 3 of 4
c
Problem 4.
c
a
a
d
b
d
e
b
e
• Using the vertices above on the right, draw the interval graph represented by the set of intervals
drawn above on the left (that is, draw the intersection graph of these intervals).
• Apply the first-fit algorithm to generate a proper coloring of the interval graph, processing the
intervals in the left-to-right order of their left endpoints. Use positive integers 1, 2, . . . as colors.
1
Fill in the colors: a:
2
b:
c:
1
3
d:
e:
2
Problem 5. An infinite sequence (a0 , a1 , a2 , . . .) has the following generating function:
f (x) =
1+x
.
(1 − x)2 (1 − 2x)
• (2 points) Compute the decomposition of f (x) into partial fractions.
f (x) =
a
b
c
1+x
=
+
+
2
2
(1 − x) (1 − 2x)
(1 − x)
1 − x 1 − 2x
1 + x = a(1 − 2x) + b(1 − x)(1 − 2x) + c(1 − x)2 = a(1 − 2x) + b(1 − 3x + 2x2 ) + c(1 − 2x + x2 )




 a+ b+ c=1
 a = −2
−2a − 3b − 2c = 1
b = −3




2b + c = 0
c=6
f (x) =
−2
−3
6
+
+
(1 − x)2 1 − x 1 − 2x
• (2 points) Using the above, write down a closed-form expression for an .
∞
∞
X
1
=
(n + 1)xn
(1 − x)2
X
1
=
xn
1−x
n=0
f (x) = −2
∞
X
n
(n + 1)x − 3
n=0
∞
X
n=0
n=0
n
x +6
∞
X
n=0
n n
2 x =
∞
X
n=0
an = −2(n + 1) − 3 + 6 · 2n
∞
X
1
=
2n xn
1 − 2x
n=0
−2(n + 1) − 3 + 6 · 2n xn
MATH 3012 H, Test 2, Answer sheet, Page 4 of 4
Problem 6. Write down the generating function of the sequence (a0 , a1 , a2 , . . .), where an is defined
below. Write your answer in as simple P
form as you can. Do not write your answer using the an s in
n
an infinite power series (in particular, ∞
n=0 an x is not an acceptable answer).
1 + x + x2
(1 − x)2
• an is the number of integer solutions to the equation
y1 + y2 + y3 = n, where y1 , y2 > 0 and 0 6 y3 6 2.
∞
Y
• an is the number of all integer partitions of n.
• an =
k=1
1
1 − xk
1
1 − 4x
4n .
n X
k + 2 n−k
• (Bonus) an =
3
.
2
1
(1 −
k=0
• (Bonus) an =
x)3 (1
− 3x)
1
√
1 − 4x
2n
.
n
Problem 7. Consider the four graphs A, B, C and D drawn below:
A
B
C
• Which of the graphs A, B, C and D are planar?
• (Bonus) Which of the graphs A, B, C and D are comparability
graphs of some posets?
• (Bonus) Which of the graphs A, B, C and D are incomparability
graphs of some posets?
D
B, D
A, C, D
A, C