1
Enumeration
1.1
Basic counting principles
1. June 2008, Question 1:
(a) How many pairs (A, B) are there with A, B ⊆ [n] and A ⊂ B? (The inclusion is
required to be strict.)
bn/2c X n
(b) Find a closed form for
.
2k
k=0
2. June 2008, Question 2:
(a) Let dn be the number of derangements of [n]. Prove that dn is odd if and only if
n is even. (By convention we define d0 =1.)
(b) Suppose that A = (aij ) is an n × n matrix with zeroes on the main diagonal such
that aij = ±1 for all i 6= j. Prove that if n is even then det(A) 6= 0.
3. January 2005, Question 1:Let dn be the number of derangements of [n] = {1, 2, . . . , n}
that is permutations π of [n] such that π(i) 6= i for all i ∈ [n].
(a) Prove that dn satisfies the recurrence relation
dn = (n − 1)(dn−1 + dn−2 )
for all n ≥ 2.
(b) By rewriting this recurrence as dn − ndn−1 = −(dn−1 − (n − 1)dn−2 ), or otherwise,
prove that
dn = ndn−1 + (−1)n .
(c) Prove that dn is even if and only if n is odd.
4. January 2008, Question 1:
(a) How many increasing functions are there which map [n] = {1, 2, . . . , n} to [m]?
[We do not require that the functions are strictly increasing.]
m X
n−k
(b) Give a closed form expression for
.
m−k
k=0
5. January 2007, Question 1:
(a) Give a combinatorial proof of the following identity:
n−1 X
n
i
=
.
k
k−1
i=k−1
[Proofs by other techniques will receive little credit.]
1
n
X
i n
.
(b) Find a closed form for the expression
i
n
i
i=1
6. June 2007, Question 1:
(a) Find a closed form for the sum
n X
i
n
i=0
2
i
.
(b) How many subsets of {1, 2, . . . , n} of size k contain no pair of elements i, j with
|i − j| ≤ 3?
n X
k
n
7. June 2006, Question 1(a): Simplify
.
m
k
k=m
8. January 2006, Question 3a: Compute
bn/2c X
k=0
n
.
2k
9. June 2004, Question 4: Give combinatorial proofs of the following facts. [Other proof
techniques will receive little credit.]
(a) s(n + 1, k) = −ns(n, k) − s(n, k − 1) where s(n, k) is the Stirling number of the
first kind, so (−1)n−k s(n, k) is the number of permutations of {1, 2, . . . , n} with
k cycles.
n
X
n
k
= n2n−1 .
(b)
k
k=1
(c) dn = (n−1)(dn−1 +dn−2 ), where dn is the number of derangements of {1, 2 . . . , n}.
10. June 2005, Question 6: Prove the following.
(a) Fn2 − Fn+1 Fn−1 = (−1)n where the Fn are the Fibonacci numbers, the solution
the recurrence Fn = Fn−1 + Fn−2 , F0 = 1, F1 = 1.
n
X
n
(b)
i
= n2n−1 .
i
i=0
.
(c) The number of solutions in integers to x1 + x2 + · · · + xk = n, xi ≥ 1 is n−1
k−1
11. January 2003, Question 1: Give combinatorial proofs of the following identities. (Other
styles of proof will receive little credit.)
(a)
(b)
k X
n+i
i=0
n
X
i=0
n+k+1
=
k
i
n
i
= n2n−1
i
2
12. January 2003, Question 2b: In order to become a bridge life master (in the American
Contract Bridge League) you need a total of 300 points, of which at least 50 must be
black, at least 50 silver, and at least 25 must be gold. The only other kind of point
available is platinum. Assume that one can only win whole numbers of each kind of
point, how many distributions of 300 points are possible which qualify one as a life
master?
13. January 2002, Question 7: Give combinatorial proofs that
(a) Dn = (n − 1)Dn−1 + (n − 1)Dn−2 and
(b) S(n, k) = kS(n − 1, k) + S(n − 1, k − 1)
where Dn is the number of derangements of {1, 2 . . . , n} and S(n, k) is the number of
partitions of {1, 2, . . . , n} into k non-empty subsets.
14. January 2002, Question 8: Determine the number of ways selecting r distinct integers
out of the first n positive integers such that the selection does not include 2 consecutive
integers.
15. June 2002, Question 2: How many solutions in integers are there to the system
(
x1 , x2 , . . . , xk ≥ 1
?
x1 + x2 + · · · + xk = m
Prove your answer. How many subsets of {1, 2 . . . , n} contain no subset of the form
{i, i + 1}, 1 ≤ i ≤ n − 1?
3
1.2
Recurrence relations
1. January 2006, Question 3b: Let cn be the number of monotonically increasing functions
from {1, 2, . . . , n} to itself with the property that f (i) ≤ i for all i. Find a recurrence
relation satisfied by the sequence (cn )∞
1 .
2. June 2005, Question 7a: Find the general form of the solution to the recurrence relation
an = an−1 + 6an−2 + 12
3. January 2005, Question 3: Solve the recurrence relation
an = −4an−1 − an−2 + 6an−3
a0 = 2, a1 = −2, a2 = 10
4. June 2004, Question 3: Solve the recurrence relation
an = 3an−2 − 2an−3
a0 = 3, a1 = 1, a2 = 8
5. June 2003, Question 3:
(a) Find the general solution the the following recurrence
an + 2an−1 − 15an−2 = 3n .
(b) A permutation π of [n] is called connected if there does not exist any i < n for
which π maps [i] to [i]. Let cn be the number of connected permutations of [n].
Prove that
n
X
ci (n − i)! = n!.
i=1
6. January 2003, Question 2(a): Find the general solution of the following recurrence
an + 5an−1 − 14an−2 = 3n .
7. January 2002, Question 5: Solve the following recurrence relation: f (n) = 2f (n − 1) +
f (n − 2) − 2f (n − 3) for n ≥ 3 where f (0) = 1, f (1) = 2, f (2) = 0.
8. June 2002, Question 1: What is the general solution to the recurrence
xn + 3xn−1 − 4xn−3 = 9?
4
1.3
Ordinary and Exponential Generating Functions, Exponential
Formula
1. Use ordinary generating functions to evaluate the sum
r
X
n
k n
.
(−1)
k
r
−
k
k=0
2. Use ordinary generating functions to evaluate the sum
2
n
X
n
k
.
k
k=0
3. Use exponential generating functions to evaluate the sum
n
X
n n−k
k−1
(−1) k
2 .
k
k=0
4. Using exponential generating functions, determine the number of n digit numbers with
all digits odd such that 1 and 3 each occur a nonzero even number of times.
5. Let gn denote the number of ways in which n students can partition themselves into
clubs that each have a president and vice president (the president and vice president
must be different students; no club can be formed with fewer than two students).
Determine the exponential generating function for hgi.
1.4
Principle of Inclusion-Exclusion
***Almost all of the following questions have the following part (a): State and prove the
Principle of Inclusion-Exclusion***
1. January 2007, Question 2: In a small town n married couples attend a town meeting,
and each of these 2n people want to speak exactly once. In how many ways can the
speakers be scheduled if we insist no married couple speaks in consecutive slots?
2. June 2007, Question 2: A class of n students walk to the park one day in single file.
On the way back the same students want to walk in single file again, but they want
to walk in an order such that no one sees the same person in front of them as they
did on the way there. In how many ways can the students be lined up satisfying this
constraint?
3. June 2006, Question 1:
n
P
k
(a) Simplify nk=m m
.
k
(b) Give, with proof, an expression for the number of surjective functions from a set
of size n to one of size k.
5
4. January 2006, Question 1: How many surjective maps are there from {1, 2, . . . , n} to
{1, 2 . . . , k}? Justify your answer.
5. June 2005, Question 10: Some married couples arrive at a dinner party. How many
different ways are there to seat them around a circular table such that no husband and
wife sit next to each other?
6. January 2005, Question 5: You are to make a necklace from n different pairs of beads.
The beads in a pair have the same colour but different shapes. In how many ways can
you make the necklace so that no two beads of the same colour are adjacent?
7. June 2004, Question 1: Prove that the number of partitions of an n-element set into k
nonempty parts satisfies
k
1 X
i k
(k − i)n .
(−1)
S(n, k) =
i
k! i=0
8. June 2003, Question 3: A derangement is a permutation π with the property that
π(i) 6= i. Prove that the number of derangements of [n] is the nearest integer to n!/e.
9. January 2003, Question 4: Give, with justification a formula for the number of surjections (onto functions) from [n] = {1, 2, . . . , n} to [k] = {1, 2, . . . , k}.
10. The chromatic function of a graph G is the function χG : N → N given by:
χG (k) = |{c : V (G) → {1, 2 . . . , k} : c is a proper vertex colouring}|.
Prove using the Principle of Inclusion - Exclusion that the chromatic function is in
fact a polynomial in k. Show moreover, that if G has n vertices and m edges then the
leading terms of χG are
χG (k) = k n − mk n−1 + . . .
1.5
Burnside’s Lemma
***All of the following questions have the following part (a): State and prove Burnside’s
Lemma concerning the number of orbits of a group action. [You may assume without proof
that if a group G acts on a set X then |Stab(x)| · |Orb(x)| = |G|, where Stab(x) is the
stabilizer of x ∈ X and Orb(x) is its orbit.]***
1. June 2008, Question 3: How many different ways are there to color the edges of K4
with two colors, red and blue? Two colorings are the same if some permutation of the
vertices takes one to the other. [One can think of this problem as counting the number
of isomorphism classes of graphs on four vertices.]
2. January 2008, Question 4: Some identity cards are to be made taking square cards
ruled into a 7 × 7 grid and punching out two of the squares. The cards can be inserted
into a scanner with any orientation. How many different identity cards can be produced
in this way?
6
3. January 2007, Question 5: How many ways are there to make a 9 bead necklace
out of red, white, and black beads if two necklaces which are rotations of each other
are considered to be the same, but necklaces which are reflections of one another are
considered distinct?
4. June 2007, Question 5: How many essentially different ways are there to color the edges
of a regular octohedron with two colors? [Octahedral dice are available on request from
the proctor.]
5. June 2006, Question 3: Compute the number of distinguishable ways there are to color
the vertices of a solid triangular prism using 3 colors.
6. January 2006, Question 4: How many distinguishable ways are there to color the faces
of an octahedral die with the colors red, white, and blue? [Octohedral dice are available
on request from the proctor.]
7. June 2005, Question 9: How many essentially different ways are there to the paint the
surface of a cube with three different colours of paint?
8. January 2005, Question 2: Some identity cards are to be made by taking square cards
ruled into a 5 × 5 grid and punching out two of the squares. How many different cards
can be produced this way?
9. June 2004, Question 2: How many ways are there to colour the squares in a Tic-TacToe grid with the colours red, white, and blue, if two colourings which differ by a
rotation or a reflection are considered the same?
10. June 2003, Question 5: How many different necklaces can e made from n beads each
of which is black or white? Two necklaces are considered the same if they only differ
by a rotation, but not if they differ by a reflection.
11. January 2003, Question 10: How many different ways are there to colour the small
triangles in the figure below with the colours red, white and blue, if rotations and
reflections count as the same colouring?
12. June 2002, Question 3: How many essentially different ways are there to paint the
corners of a solid cube with 5 colours?
7
1.6
The basic probabilistic method and linearity of expectation
1. (a) Use a random partition of the vertices to prove that every graph has a bipartite
subgraph with at least half of its edges.
(b) Prove that every graph G with m edges that has a matching with k edges has a
bipartite subgraph with at least (m + k)/2 edges.
2. An instance of SATISFIABILITY is a list of clauses where each clause is a set of literals
(a literal is a variable or its negation). A satisfying assignment sets each variable TRUE
or FALSE so that each clause has at least one true literal. Given that all clauses have
size k, prove that the minimum number of clauses in an unsatisfiable instance is 2k .
3. A tournament is an orientation of Kn . A subset of vertices
tournament is beaten
in a−k
n
if some vertex beats every element of it. Prove that if k (1 − 2 )n−k < 1 then there
exists a tournament on n vertices such that every k-set is beaten.
4. If G is a graph then
α(G) ≥
X
v∈V
1
.
d(v) + 1
(Hint: Pick a random ordering on the vertices.)
8
2
Discrete Structures, Coding Theory, Design Theory,
Posets
2.1
Coding Theory Basics
1. January 2006, Question 2: Let C be a binary code of length n with minimum distance
d > n/2 having M codewords. Enumerate C as C = {c1 , c2 , . . . , cM }.
(a) Prove that
M (M − 1)d ≤
M X
M
X
i=1 j=1
d(ci , cj ) ≤
nM 2
.
2
(b) Deduce that C has at most 2d/(2d − n) codewords. (This bound is called the
Plotkin bound.)
2.2
Linear Codes
1. January 2008, Question 5: Given integers 0 < d ≤ n and a prime power q prove that
there exists a linear code C ⊆ Fnq of minimum distance at least d, containing at least
!
X
d−1 n
qn
(q − 1)i
i
i=0
codewords.
2. January 2007, Question 4:
(a) Let C be a binary linear code and let E be the subset of C consisting of those
codewords having even length Prove that |E| is either |C| or |C|/2.
(b) Prove that given positive integers n, q, d there is a q-ary code of lenght N and
minimum distance d having at least
qn
Pd−1 n
i
i=0 i (q − 1)
codewords. [Note: we are not requiring this code to be linear.]
2.3
Sphere packing
1. June 2004, Question 5:
(a) State and prove the Hamming (or “sphere packing” ) bound on the number of
words in a q-ary code with length n and minimum distance at least d.
(b) Prove that for binary codes the Hamming bound is always at least as strong as
the Singleton bound.
2.4
Generalized Reed Solomon Codes
9
2.5
Posets
1. January 2008, Question 3: Suppose 0 < t < n/2 and that A ⊂ P(n) is an antichain
with |A| ≤ t for all A ∈ A. Define
[n]
At = B ∈
: there is some A ∈ A with A ⊆ B .
t
Prove that |A| ≤ |At |.
2. January 2007, Question 3:
(a) Prove that if P is a poset such that no chain has length more than k then P can
be written as the union of at most k antichains. [Hint: This is not Dilworth’s
theorem.]
(b) Prove that a sequence of distinct real numbers of length n2 must contain either
an increasing subsequence of length n + 1 or a decreasing subsequence of length
n + 1.
3. June 2007, Question 3: State and prove Sperner’s lemma concerning the largest size
of an antichain in the power set of {1, 2, . . . , n}.
P
4. June 2006, Question 4: Let c1 , c2 , . . . , cn P
≥ 0 be real numbers such that ni=1 ci = 1.
Prove that the collection A = {A ⊂ [n] : i∈A ci > 21 } has size at most 2n−1 .
5. January 2006, Question 2: State and prove the Erdos-Ko-Rado Theorem concerning
the maximum size of an intersecting family of k-sets.
6. January 2002, Question 6: A family F of subsets of X is intersecting if A, B ∈ F ⇒
A ∩ B 6= ∅.
(a) Prove that an intersecting family F of subsets of X = {1, 2 . . . , n} satisfies |F| ≤
2n−1 .
(b) Prove that any intersecting family F of X = {1, 2, . . . , n} can be extended to an
intersecting family of size 2n−1 .
7. Prove that a poset of size greater than mn has a chain of size greater than m or an
antichain of size greater than n. Use this to prove the Erdos-Szekeres Theorem: every
list of mn + 1 distinct integers has an increasing sublist with more than m elements or
a decreasing sublist with more than n elements.
8. A family of sets is union free if it does not have two distinct members whose union
is a third member of the family. Moser asked for the maximum size of a union-free
subfamily that can be guaranteed to exist in any family of n sets. Use Dilworth’s
Theorem to give a short
√ proof that every family of n distinct sets contains a union-free
family of size at least n.
9. Prove that the largest
antichain of subsets of [n] consisting of pairs of complementary
n−1
sets has size 2 dn/2e
. (Hint: Use the Erdos-Ko-Rado Theorem.)
10
10. Prove the following dual of Dilworth’s Theorem: If P is a finite poset, then the maximum size of a chain in P equals the minimum number of antichains needed to cover
the elements of P . (Hint: The set of maximal elements is an antichain. Remember
that there are two inequalities needed to be shown.)
11. Let n ≤ 2k and let A1 , . . . , Am be a family of k-element subsets of [n] such that
Ai ∪ Aj 6= [n] for all i, j. Show that m ≤ (1 − nk ) nk . (Hint: Apply the Erdos-Ko-Rado
theorem to the complements Ai .)
12. Let F be a k-uniform family, and suppose that it is intersection free, i.e., that A∩B 6⊂ C
k
for any three sets A, B, and C of F. Prove that |F| ≤ 1 + bk/2c
. (Hint: Fix a set
B0 ∈ F and observe that {A ∩ B0 : A ∈ F, A 6= B0 } is an antichain over B0 .)
13. Let x1 , . . . , xn be real numbers, xi ≥ 1 for each i and let S be the set of all numbers
which can be obtained as linear combinations α1 x1 + · · · + αn xn with αi ∈ {−1, +1}.
Let I = [a, b) be any interval (in the real line) of length b − a = 2. Show that
n
|I ∩ S| ≤ bn/2c
.
2.6
Designs
1. June 2003, Question 2 & January 2003, Question 3 & January 1995, Question 9: Let
B = {B1 , B2 , . . . , Bb } be a family of subsets (called blocks) of X = [v]. Further, assume
that each unordered pair {j, m} ⊂ X occurs in exactly λ > 0 blocks. Prove that if
each i ∈ X belongs to strictly more than λ of the blocks then b ≥ v.
2. January 2002, Question 9: Let B = {B1 , B2 , . . . , Bb } be a family of subsets (called
blocks) of a v-set X = {x1 , x2 , . . . , xv }. Further, assume that each unordered pair
{xj , xm }, 1 ≤ j < m ≤ v, occurs in exactly λ > 0 blocks. Prove that if λ < |Bi | < v,
for 1 ≤ i ≤ v, then b ≥ v. (Hint: Use an incidence matrix and determinant.)
3. In a block design, block sizes can vary. Construct blocks (subsets of [v] for some v)
such that any two elements of [v] appear together in one block, all elements appear
equally often, and the block sizes are not all equal.
4. A Steiner quadruple system (SQS) is a collection Q of 4-element subsets of X such that
for any 3-subset T of X there exists a unique Q ∈ Q such that T ⊆ Q. The integer
n = |X| is called the order of Q.
(a) Prove that if Q is an SQS of order n, then |Q| = n(n − 1)(n − 2)/24.
(b) Let V be a (finite dimensional) vector space over F2 . Define
(
)
X
V
Q= Q∈
:
v=0 .
4
v∈Q
Prove that Q is an SQS on the ground set V .
11
5. When q ≥ 2 a family of (q + 1)-sets is the family of lines of a projective plane of order
q if and only if the family is a symmetric (q 2 + q + 1, q + 1, 1) design.
6. Let B be a balanced incomplete block design with parameters for v, k, λ, b, r whose set
of varieties is V = {x1 , . . . , xv } and whose blocks are B{B1 , . . . , Bb }. For each block
Bi , let Bi = V \ Bi . Let B c denote the collection of subsets B 1 , . . . , B b of V . Prove
that B c is a block design if b − 2r + λ > 0, and determine its parameters v 0 , k 0 , λ0 , b0 , r0 .
12
3
Graph Theory
3.1
Basic Concepts of Graphs
1. January 2007, Question 6: Recall that a trail is a walk in a graph that does not use
any edge more than once. A graph is called randomly Eulerian from vertex x if every
maximal trail starting at x is an Euler circuit. Prove that G is randomly Eulerian
starting at x if and only if G has an Euler circuit and x is contained in every cycle of
G.
2. June 2006, Question 6: Prove that every non-trivial tree contains at least two maximal
independent sets, with equality only for stars.
3. January 2006, Question 1:
(a) Prove that a connected graph on n vertices has n edges if and only if it contains
exactly one cycle.
(b) Let n ≥ 3 and suppose that G is an n vertex graph with the property that for all
v ∈ V (G) the graph G \ {v} is a tree. Determine the number of edges in G, and
thereby determine G.
4. June 2005, Question 1: [Note that in both parts we are counting different graphs, not
non-isomorphic graphs.]
n
(a) Prove that the number of graphs with vertex set {1, 2, . . . , n} is 2( 2 ) .
(b) Prove that the number of graphs with vertex set {1, 2, . . . , n} such that all the
n−1
vertex degrees are even is 2( 2 ) .
5. January 2005, Question 7: Let d = d(G) = 2e(G)/n(G) > 0 be the average degree of
a graph G.
(a) Prove that G has a subgraph H with δ(H) > d/2. [Hint: consider successively
deleting vertices of degree at most d/2.]
(b) Prove that for all c < 1/2 there exists a graph G with no subgraph H having
δ(H) > cd(G).
[Hint: Consider K1,n .]
6. June 2004, Question 8: Prove that every graph has a bipartition V (G) = X ∪ Y with
the property that e(X, Y ) ≥ e(G)/2. Further, show that if G is 3-regular then we can
achieve e(X, Y ) ≥ n(G) = 2e(G)/3.
7. June 2003, Question 6:
(a) Suppose that G is a k-regular graph with k ≥ 1, having n vertices. Prove that
α(G) ≤ n/2.
13
(b) Suppose that T is a tree having n vertices. Prove that α(T ) ≥ n/2 with equality
if and only if T has a perfect matching.
8. January 2003, Question 6: Let T be a tree. We say that T splits oddly if for every edge
e ∈ E(T ) the two components of T − e each have an odd number of vertices. Prove
that the vertices of T all have odd degree if and only if T splits oddly.
9. January 2002, Question 3:
(a) State and prove a necessary and sufficient condition for a connected graph G to
be Eulerian in terms of its degrees.
(b) By finding an Euler circuit in a suitably defined directed graph, construct a circular binary sequence S such that each binary word of length 4 appears exactly
once as one moves along the sequence S.
10. June 2002, Question 9: Let G be a graph with vertex set V = {v1 , v2 , . . . , vn }. The
adjacency matrix of G is the n × n matrix A = [aij ] with
(
1 vi is adjacent to vj in G
aij =
.
0 otherwise
Prove that the ij th entry of Ak is the number of walks of length k in G from vi to vj .
11. June 2002, Question 10: Let G be a connected bipartite graph which is k-regular for
some k ≥ 2. Prove that G is bridgeless.
12. June 2007, Question 6: Consider a family A = {A1 , A2 , . . . , An } of distinct subsets of
some n-set X. Define a graph G on A with an edge between Ai and Aj if |Ai ∆Aj | = 1.
If Ai ∆Aj = {x} then we label the edge Ai Aj with x. Prove that there is a forest F ⊂ G
whose edges include all the labels used on edges of G. Deduce that there exists some
x ∈ X for which the sets A1 \ {x}, A2 \ {x}, . . . , An \ {x} are distinct.
13. June 2007, Question 10: In this question we consider directed graphs (digraphs) with
no loops and no multiple edges (but we do allow both x → y and y → x). A monotone
tournament is an orientation of a complete graph in which (for some ordering of the
vertices) the ordering on the edges is from the smaller to the larger vertex. A complete
digraph is a digraph in which both x → y and y → x are edges for every x, y in its
vertex set.
Given m ≥ 1 prove that if N is sufficiently large then every digraph on N vertices
contains a subset of size m which induces either an empty digraph, a complete digraph,
or a monotone tournament.
3.2
Matchings
1. June 2008, Question 6: Prove the Konig-Egerváry Theorem from the vertex version of
Menger’s Theorem.
14
2. June 2008, Question 7(a): Prove that every tree has at most one perfect matching.
3. June 2007, Question 9: Prove that a tree T has a perfect matching if and only if the
number of odd components in G − v is 1 for every vertex v of G.
4. June 2006, Question 2: Let G be a bipartite graph with bipartition A, B that contains
a matching from A to B.
(a) Prove that for some vertex a ∈ A it is the case that for all edges ab ∈ E(G) there
is a matching from A to B that contains ab. [Hint: It may be helpful to consider
whether or not there is subset C ⊂ A having exactly |C| neighbours.]
(b) Deduce that if all vertices in A have degree d then the number of matchings from
A to B in G is at least d! if d ≤ |A| and at least d(d − 1)(d − 2) . . . (d − |A| + 1)
if d > |A|.
m
5. January 2006, Question 4: Suppose that X is a set of size mn and (Ai )m
1 , (Bi )1 are
partitions of X into m sets of size n. Prove that one can renumber the sets Bi in such
a way that Ai ∩ Bi 6= ∅ for i = 1, 2 . . . , m.
6. January 2006, Question 5:
(a) State Tutte’s Theorem concerning graphs having a 1-factor.
(b) Prove that every connected, bridgeless, 3-regular graph has a 1-factor.
(c) Find a connected 3-regular graph with no 1-factor.
7. June 2004, Question 6: Let A1 , A2 , . . . , An be finite sets and d1 , d2 , . . . , dn non-negative
integers. Prove that there are disjoint subsets Di ⊂ Ai with |Di | = di if and only if for
all I ⊂ {1, 2 . . . , n} we have
[ X
di .
Ai ≥
i∈I
i∈I
8. June 2003, Question 4:
(a) State Hall’s theorem concerning systems of distinct representatives.
(b) A positional game consists of a set X of positions and a set W ⊂ P(X) of winning
sets of positions. [For instance in Tic-Tac-Toe the positions are X = {1, 2, 3}2
and the wining sets are those which contain a line.] Two players alternately select
positions from X until one player’s set of selected positions is in W. (No position
can be selected twice.) Suppose that |W | ≥ a for all W ∈ W and no x ∈ X
belongs to more than b winning sets. Prove that the second player can force a
draw if a ≥ 2b. [Hint: consider the bipartite graph with vertex classes X and two
disjoint copies of W. Join x ∈ X to W ∈ W whenever x ∈ W.]
9. January 2003, Question 5:
(a) State Hall’s theorem concerning systems of distinct representatives.
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(b) A permutation matrix is {0, 1}-matrix having exactly one 1 in every row and
column. Prove that an n × n matrix A with non-negative integer entries is a sum
of permutation matrices if and only if all the row and column sums are equal.
10. June 2002, Question 6: Suppose that G is a bipartite graph with bipartition (X, Y ).
Define the deficiency of a subset S ⊂ X to be def(S) = |S| − |N (S)|. Prove that the
maximum size of a matching in G is
|X| − max{def(S) : S ⊆ X}.
3.3
Connectivity
1. June 2007, Question 7: Prove that tree T having 2k endvertices contains k edge-disjoint
paths joining all the endvertices in pairs. Deduce if x is a vertex in a graph G with
degree 2k and x is not a cutvertex of G then x is contained in k edge-disjoint cycles.
2. June 2006, Question 7:
(a) State the Max Flow/Min Cut theorem.
(b) Using the Max Flow/Min Cut theorem or otherwise prove that given a collection
(Ai )n1 of sets and integers (di )n1 one can find di distinct representatives for Ai if
and only if for all S ⊂ {1, 2 . . . , n} we have
[ X
di .
Ai ≥
i∈S
i∈S
[To be precise we want to find distinct elements xij for 1 ≤ i ≤ n, 1 ≤ j ≤ di with
xij ∈ Ai .]
3. June 2006, Question 10: Suppose that G = G(n) is a graph such that no two vertices
of G are joined by 3 internally vertex disjoint paths. Prove that e(G) ≤ 23 (n − 1).
[Hint: Consider the block cut-vertex graph of G.]
4. January 2006, Question 2: Prove that if G is a k-connected graph and S, T ⊂ V (G)
are disjoint subsets of vertices, each of size k then it is possible to find k disjoint paths
P1 , P2 , . . . , Pk in G and labelings S = {s1 , s2 , . . . , sk }, T = {t1 , t2 , . . . , tk } such that Pi
is a path from si to ti .
5. June 2005, Question 3: Let k ≥ 1. Prove that if G is k-connected and S is a set of k
vertices of G then there is a cycle C ⊂ G with S ⊆ V (C). [Hint: you may assume the
Fan Lemma provided you state it clearly.]
6. January 2005, Question 6: Let k be an integer with k ≥ 1, let G be a k-connected
graph, and let S, T ⊂ V (G) be disjoint sets of vertices of G, each of size at least k.
Prove that G contains k disjoint S, T paths. (No version of the Fan Lemma may be
assumed without proof.)
7. January 2003, Question 9:
16
(a) Define a block of a graph G, and the block-cutvertex tree associated with G.
(b) Prove that if G has blocks B1 , B2 , . . . , Bk then
n(G) = 1 − k +
k
X
n(Bi ).
i=1
(c) Obtain, with justification, a formula for the number of spanning trees of G, given
that the number of spanning trees of Bi is ti .
3.4
Coloring, Turán’s Theorem
1. June 2008, Queston 7(b): Let G be a graph with 2m + 1 vertices and more than
m · ∆(G) edges. Prove that χ0 (G) > ∆(G).
2. June 2008, Question 8: Prove that χ(G) = ω(G) when the complement of G is bipartite.
3. June 2008, Question 10: [An alternative proof of Turán’s theorem.] Let G be a maximal
Kr -free graph (maximal in the sense of edges) on at least r − 1 vertices.
(a) Prove that G contains a Kr−1 .
(b) Show that if A is a clique in G of size r − 1 then
e(G) ≤ e(Kr−1 ) + (n − r + 1)(r − 2) + e(G \ A).
(c) Give a proof of Turán’s theorem using the previous part.
4. January 2007, Question 8: Let G be an edge-maximal graph on n vertices not containing a Kr and suppose that n ≥ r + 1. Let Tr−1 (n) be the complete (r − 1)-partite graph
on n vertices whose class sizes are as equal as possible, and let tr−1 (n) = e(Tr−1 (n)).
(a) Prove that G contains a Kr−1 on some subset A ⊂ V (G).
(b) Prove that e(G) ≤ r−1
+ (r − 2)(n − r + 1) + e(G \ A) and deduce that e(G) ≤
2
tr−1 (n).
(c) Prove that a graph on n vertices having tr−1 (n) edges and not containing a Kr is
Tr−1 (n).
5. January 2007, Question 9: An interval graph is a graph whose vertices consist of
closed intervals in R, and where two intervals are adjacent if they are not disjoint.
Prove that if G is an interval graph then χ(G) = ω(G). [Hint: consider using the
greedy algorithm.]
6. June 2007, Question 8: Let x be a vertex of a graph G. For r ≥ 0 define
Gr = G[{y ∈ V (G) : dG (x, y) = r}].
(In other words, Gr is the subgraph of G induced by the vertices at distance r from
x.) Prove that
χ(G) ≤ maxr (χ(Gr ) + χ(Gr+1 )).
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7. June 2006, Question 5: Let
1
1
1
+ 1.
f (k) = k! 1 + + + · · · +
2! 3!
k!
k 3’s
z }| {
Prove that R(3, 3 . . . , 3 ≤ f (k), or in other words that if you color the edges of Kn
with k colors and n ≥ f (k) then the coloring contains a monochromatic triangle.
8. June 2006, Question 8: State and prove Turán’s theorem, including a proof that the
extremal graph is unique up to isomorphism.
9. June 2006, Question 9: Given an orientation of a graph G (an assignment of a direction
to each edge of G) we define the length of the orientation to be the length of the longest
directed path in G.
(a) Prove that if χ(G) ≤ k then G has an orientation of length at most k.
(b) Prove the converse: that if G has an orientation of length at most k then χ(G) ≤ k.
10. June 2005, Question 2: Let G be a graph that does not contain two disjoint odd cycles.
Prove that χ(G) ≤ 5. Exhibit such a graph with χ(G) = 5.
11. June 2005, Question 4 & January 2003, Question 7: State and prove Turán’s theorem
concerning the maximum number of edges in a graph on n vertices not containing a
Kr .
12. January 2005, Question 8: Prove that χ(G) = ω(G) when G is bipartite.
13. January 2005, Question 9 & June 2004, Question 10:
(a) State Turán’s theorem concerning the maximum number of edges in a graph on
n vertices containing a Kr .
(b) Prove that if G is a graph with n ≥ r + 1 vertices and tr−1 (n) + 1 edges then for
every n0 with r ≤ n0 ≤ n there is a subgraph H of G with n0 vertices and at least
tr−1 (n0 ) + 1 edges. [Hint: consider a vertex in G of minimum degree.]
(c) From the previous part deduce Turán’s theorem, and also the stronger fact that
such a G contains two Kr subgraphs sharing r − 1 vertices.
14. June 2003, Question 10: Define R(s, t) to be the minimum value of n such that every
(non-necessarily) proper edge colouring of Kn with the colours red and blue contains
a monochormatic red Ks or a monchromatic blue Kt . prove that for all s, t ≥ 2,
R(s, t) ≤ R(s − 1, t) + R(s, t − 1)
and deduce that for all such s, t
s+t−2
R(s, t) ≤
.
s−1
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15. June 2003, Question 8: State and prove Turán’s Theorem concerning graphs not containing a copy of Kr .
16. January 2003, Question 8:
(a) Prove the Szekeres-Wilf Theorem, stating that χ(G) ≤ 1 + maxH⊂G δ(H).
(b) Given a set of N lines in the plane in general position (no two parallel, no three
meeting at a point) define a graph on the vertex set consisting of the intersection
points of the lines, with two vertices adjacent if they appear consecutively on one
of the liens. Prove that the chromatic number of this graph is at most 3.
17. June 2002, Question 7: Let G be a graph containing no induced subgraph isomorphic
to P4 . Prove that given any ordering of the vertices of G, the greedy algorithm colours
G in ω(G) colours. [Hint: If the greedy algorithm uses k colours, consider the smallest
i such that G contains a clique consisting of vertices coloured i, i + 1, i2 , . . . , k}.]
3.5
Planar Graphs
1. June 2008, Question 9: Prove that a set of edges in a connected plane graph G forms
a spanning tree of G if and only if the duals of the remaining edges form a spanning
tree of the dual graph G∗ .
2. January 2006, Question 3: Prove that every planar graph G has χ(G) ≤ 5.
3. June 2005, Question 5:
(a) State Euler’s formula concerning the number of faces, edges, and vertices of a
plane graph.
(b) Prove that a plane graph with n vertices and e edges has e ≤ 3n − 6 provided
n ≥ 3.
(c) An outerplane graph is a plane graph in which all the vertices are on the boundary of the outer face. What is the maximum number of edges in an outerplane
graph with n vertices? Justify your answer.
4. January 2005, Question 10: Consider a connected plane graph G with dual G∗ . Prove
that a subset of E(G) forms a spanning tree if and only if the duals of the remaining
edges form a spanning tree of G∗ .
5. January 2002, Question 2:
(a) Define the term “maximal planar graph.”
(b) Prove that if G is a maximal planar graph with p ≥ 3 vertices and q edges, then
q ≤ 3p − 6.
(c) Prove that there exists only one 4-regular maximal planar graph.
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6. June 2002, Question 8: State and prove Euler’s Formula concerning planer graphs.
The girth of a graph G is the length of the shortest cycle in G. Prove that if G is
planar and bridgeless with n vertices, m edges, and girth g, then
m≤
3.6
g
(n − 2).
g−2
Hamiltonian Cycles
1. January 2007, Question 10:
(a) Let G be a graph on n ≥ 3 vertices such that for all pairs of non-adjacent vertices
x, y we have d(x) + d(y) ≥ n. Prove that G has a Hamilton cycle. [Hing: consider
a longest path in G.]
(b) Prove that the result in part (a) is best possible, by constructing a graph for
every n ≥ 3, have n vertices and satisfying d(x) + d(y) ≥ n − 1 for all pairs of
non-adjacent vertices which does not have a Hamilton cycle.
2. January 2002, Question 1:
(a) Show that if a graph is not connected, then its complement is connected.
(b) A graph G has a proper edge coloring with k colors if no two edges of the same
color meet at a common vertex. Show that if G is a regular graph with degree 3,
where G is Hamiltonian, then G has a proper edge coloring with three colors.
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