Combinatorics Qualifying Exam
August, 2016
This examination consists of two parts, Combinatorics and Graph Theory. Each part
contains five problems of which you must select three to do. Each problem is worth 20
points. Only hand in your solutions to three problems from each part. Please do not turn
in more solutions since only the first three solutions from each part will be graded.
Begin each problem on a new sheet of paper and be sure to label each page of your work
with the problem number and your name.
In each question, if you appeal to a theorem within your solution, you must carefully
state the entire theorem. All graphs, unless otherwise stated, should be understood to be
finite and simple.
Arithmetic expressions need not be completely reduced unless otherwise stated, and
may be left in terms of sums, differences, products, quotients, exponentials, factorials, and
binomial coefficients.
1
Combinatorics. Do any three. In all questions below, [n] represents the set
{1, 2, 3, . . . , n}.
Problem A1: Let the set S = [n] × [n] be partially ordered by the relation (a, b) (c, d)
which holds true when a ≤ c and b ≤ d. Find, with proof, the width and height of this poset.
Problem A2: Prove that for any positive numbers m and n,
n X
n
k=0
k
S(k, m) = S(n + 1, m + 1)
where S(·, ·) is the Stirling number of the second kind.
Problem A3: Let an represent the number of strings of the letters “A”, “B”, “C”, and “D”
such that the letter “A” appears an odd number of times (e.g. a0 = 0, a1 = 1, and a2 = 6).
Find a closed formula for an .
Problem A4: Let π and σ be elements of Sn2 +1 . Prove that there is either a set A of n
elements of [n2 + 1] such that for all i, j ∈ A, π(i) < π(j) if and only if σ(i) < σ(j), or that
there is a set B of n elements of [n2 + 1] such that for all i, j ∈ B, π(i) < π(j) if and only if
σi > σj
Problem A5: Prove that from any set of 10 positive integers between 10 and 99 inclusive,
it is always possible to select two disjoint nonempty subsets whose elements have the same
sum.
2
Graph Theory. Do any three.
Problem B1: Prove that every graph G on 2n vertices with minimum degree δ(G) ≥ n
contains a perfect matching.
Problem B2: Prove √
that if G is a graph on n vertices with complement G, it is the case
that χ(G) + χ(G) ≥ 2 n.
Problem B3: Let G be a cubic graph with edge-chromatic number 3 such that the partition
of E(G) into three color classes is unique. Prove that G is Hamiltonian.
Problem B4: Recall that the diameter of a graph G is d(G) = maxu,v∈G d(u, v). Prove
that a tree T contains a vertex v from which every other vertex is at a distance of at most
)
e.
d d(T
2
Problem B5: Prove that if G is a graph with at least 11 vertices, then either G or its
complement G is nonplanar.
3
Combinatorics Qualifying Exam
January 5, 2015
This examination consists of two parts, Combinatorics and Graph Theory.
Each part contains five problems of which you must select three to do. Each
problem is worth 20 points. Only hand in your solutions to three problems
from each part. Please do not turn in more solutions since only the first three
solutions from each part will be graded.
Begin each problem on a new sheet of paper and be sure to label each
page of your work with the problem number and your name.
In each question, if you appeal to a theorem within your solution, you
must carefully state the entire theorem. All graphs, unless otherwise stated,
should be understood to be finite and simple.
In the unlikely case an exercise requires you to prove a false statement,
provide a proof that the statement is false.
1
Combinatorics
Do any three.
1. Let d > 0 be an integer, and 0 < p < 1. Show that for a sufficiently
large integer n,
d X
n n−d
n−d
n−m
n
d−m
d!pd
(d − m)!p
<2
d
d
d
−
m
d
−
m
m
m=0
2. A non-crossing pairing of the set N = {1, . . . , 2n} is a partition of N
into 2-element subsets such that if {i, i′ } is a part with i < i′ , and
{j, j ′ } is a part with j < j ′ , then it is not the case that i < j < i′ < j ′ .
How many non-crossing pairings are there?
3. Recalling
P thatn S(n, k) is the Stirling number of the second kind, prove
S(n − m, k) = (k + 1)S(n, k + 1) for all nonnegative
that nm=1 m
integers n and k.
4. Prove that for every set of n integers there is a nonempty subset that
sums to a multiple of n.
5. Let an be the number of strings (or words) of length n consisting of the
letters A, B, C, and D which contain at least one A and at least one
D. Find a closed formula for an .
2
Graph Theory
Do any three.
1. Let G be a graph. We say a set of vertices S forms a generalized
cycle if S is the vertex set of a cycle, or a path on a single edge,
or |S| = 1. Show that V (G) can be partitioned into at most α(G)
generalized cycles. (Hint: start with a maximal path.)
2. Prove that every two-coloring of the edges of Kn contains a monochromatic spanning tree.
3. Suppose G has at least 5 vertices and that every induced subgraph on 3
vertices has the same number of edges. Show that G is either complete
or empty.
4. Let G be a bipartite graph with partite sets A, B. Suppose |A| = a,
and |B| = a + b for a, b positive integers. Suppose that there is a
subset Y ⊆ B such that |N(Y )| < |Y | − b. Show that G does not have
a matching that covers every vertex of A.
5. Let G be a simple finite graph with edges e1 , e2 , e3 such that G −
{e1 , e2 , e3 } has no cycles. Prove that G is planar.
3
Combinatorics Qualifying Exam
August, 2014
This examination consists of two parts, Combinatorics and Graph Theory. Each part
contains five problems of which you must select three to do. Each problem is worth 20
points. Only hand in your solutions to three problems from each part. Please do not turn
in more solutions since only the first three solutions from each part will be graded.
Begin each problem on a new sheet of paper and be sure to label each page of your work
with the problem number and your name.
In each question, if you appeal to a theorem within your solution, you must carefully
state the entire theorem. All graphs, unless otherwise stated, should be understood to be
finite and simple.
Arithmetic expressions need not be completely reduced unless otherwise stated, and
may be left in terms of sums, differences, products, quotients, exponentials, factorials, and
binomial coefficients.
1
Combinatorics. Do any three.
Problem A1: Prove that for nonnegative integer n and integer r ≥ 2:
n 2
X
n
i=0
i
n X
n 2n − j
r =
(r − 1)j
j
n
j=0
i
Problem A2: Prove that for any positive integer n there are the same number of partitions
of n into any number of parts none of which are divisible by 3 as there are partitions of n
into any number of parts such that the same number does not appear more than twice in
the partition.
Problem A3: Let k be an arbitrary positive integer. Prove that k has a positive multiple
in which all decimal digits are either 0 or 1. (Hint: the Pigeonhole Principle may be useful.)
Problem A4: How many different ways are there to paint the squares of a 3×3 checkerboard
with 3 colors, if each color must be used at least once and two boards are considered to be
identical if one can be flipped or rotated to get the other?
Problem A5: (a) Show that every finite poset can be embedded into a hypercube of
sufficiently large dimension.
(b) Let h(P ) denote the least positive integer n such that P can be embedded into an
n-dimensional hypercube. Let Ak denote the k-element antichain. Find h(A12 ).
Please recall: the n-dimensional hypercube is the poset on the power set of {1, 2, 3, . . . , n}
ordered by the subset relation, and that an embedding of one poset into another is an injective
function f such that x ≤ y if and only if f (x) ≤ f (y).
2
Graph Theory. Do any three.
Problem B1: Prove that a graph with n vertices and independence number α contains a
path on at least d αn e vertices (note that a single vertex may be considered to be a degenerate
path on 1 vertex).
Problem B2: Let T be a tree with an even number 2k of leaves. Prove that it is possible
to label the leaves u1 , u2 , . . . , uk and v1 , v2 , . . . , vk such that all k of the unique paths from
ui to vi have a vertex in common.
Problem B3: A graph is greedy-set colored by the following method: we repeatedly select
the largest independent set of uncolored vertices (choosing an arbitrary set if there are
several) and assign it a new color. We repeat this procedure until all vertices are colored.
Determine whether there is an absolute constant k such that this algorithm uses at most
χ(G) + k colors.
Problem B4: Prove that there is a coloring of the edges of a K6,6 with two colors such
that there is not a monochromatic K3,3 .
Problem B5: Prove that for k ≥ 2, for any k vertices of a k-connected graph, there is a
cycle passing through all of them.
3
Combinatorics Qualifying Exam
August, 2012
This examination consists of two parts, Combinatorics and Graph Theory. Each part
contains five problems of which you must select three to do. Each problem is worth 20
points. Only hand in your solutions to three problems from each part. Please do not turn
in more solutions since only the first three solutions from each part will be graded
Begin each problem on a new sheet of paper and be sure to label each page of your work
with the problem number and your name.
In each question, if you appeal to a theorem within your solution, you must carefully
state the entire theorem. All graphs, unless otherwise stated, should be understood to be
finite and simple.
1
Combinatorics. Do any three.
Problem A1: Let S = {1, 2, . . . , n}. A subset A of S is called a triple if |A| = 3. The
triples A and B are independent if |A ∩ B| ≤ 1. Prove that if F = {A1 , A2 , . . . , Am } is a
collection of pairwise independent triples from S, then
m≤
n(n − 1)
.
6
Problem A2: Determine the number of ways a 2 × n chessboard can be covered with 1 × 2
and 2 × 2 rectangles. The chessboard is considered to be unrotatable, but the dominoes may
be placed in either a 1 × 2 or 2 × 1 orientation.
Problem A3: Find a formula for the number of ways the faces of a dodecahedron can be
colored with n different colors, if each color must be used at least once and if two colorings
are considered to be equivalent if one is a rotation of the other. (Hint: there are 60 ways to
rotate of a dodecahedron onto itself,
Problem A4: For a given partition q of the integer n, let t(n, q) be the number of copies
of the number 2 appearing
in q, and let u(n,
P q) be the number of parts which appear exactly
P
once in q. Let an = q t(n, q) and bn = q u(n, q). For example, 5 may be partitioned into
5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1. These respectively
have t(5, q) values of 0, 0, 1, 0, 2, 1, and 0, for a total of a5 = 4, while the values of u(5, q)
are respectively 1, 2, 2, 1, 1, 1, and 0, so b5 = 8. Prove that an = bn−1 for all n > 0. (Hint:
let P (x) be the generating function for the number of partitions of n, and let A(x) and B(x)
be the generating functions for an and bn respectively, and find formulas for A(x) and B(x)
in terms of P (x).)
Problem A5: For integer n > 0, evaluate the sum
2
Pn
k=0
k2
n
k
.
Graph Theory. Do any three.
Problem B1: Prove that if G contains at least one cycle and has girth at least 5, then the
complement of G is Hamiltonian.
Problem B2: Determine, with proof, a formula for ex(n, P4 ), where P4 is the path on four
vertices, and ex represents the extremal number.
Problem B3: Prove that every 4-connected graph contains a subdivision of K4 .
Problem B4: Prove that for a graph G of independence number α(G), there is a vertexdisjoint union of α(G) (not necessarily induced) subgraphs of G, each of which is a cycle, an
isolated vertex, or a K2 , which covers every vertex of G.
Problem B5: Prove that for all n ≥ 2 there exists a graph G on 2n vertices that has the
following property: for all disjoint A, B ⊆ V (G) with |A| = |B| = 2n, there are vertices
a ∈ A and b ∈ B such that ab ∈ E(G), and there are also vertices a0 ∈ A and b0 ∈ B such
that a0 b0 6∈ E(G).
3
Combinatorics Qualifying Exam
January 7, 2011
This examination consists of two parts, A and B. Part A contains six problems
of which you must select four to do. Part B contains three problems of which you
must select two to do. Each problem in part A is worth 15 points and each problem
in part B is worth 20 points. Only hand-in your solutions to four problems from part
A and two from part B. Please do not turn-in more solutions since only the first four
solutions from part A will be graded and only the first two solutions from part B will
be graded.
Begin each problem on a new sheet of paper and be sure to label each page of
your work with the problem number and your name.
In each question, if you appeal to a theorem within your solution, you must
carefully state the entire theorem. All graphs, unless otherwise stated, should be
understood to be finite and simple.
1
Part A: (15 points each) Submit solutions to any four.
Time: 2 hours.
Problem A1: Let us call a set of integers pairwise divisible if for any two elements of
the set, one is divisible by the other, and pairwise indivisible if for any two elements,
neither is divisible by the other.
a) Find a set of 9 numbers such that no four of them are pairwise divisible and
no four of them are pairwise indivisible.
b) Prove that a set of 10 numbers contains either four pairwise divisible numbers or four pairwise indivisible numbers.
Problem A2: Find the number of ways of arranging the 26 letters of the alphabet
so that no one of the sequences ABC, P QRS, and XY Z appears.
Problem A3: A 1 × n checkerboard is to be covered with the following objects:
pennies, nickels, and dimes, each of which covers one square, and dominoes, which
cover two squares. Let bn be the number of possible configurations of objects so that
every square is covered.
a) Find a recurrence relation describing bn .
b) Find a closed formula for bn .
Problem A4: Prove that every spanning subgraph of the complete bipartite graph
Kn,n with minimum degree at least n/2 has a perfect matching.
Problem A5: Let G be a simple plane graph such that every face is a triangle. Show
that χ(G) = 3 if and only if G has an Eulerian circuit.
Problem A6: Prove that if G is a graph with n vertices, then χ(G)χ(Gc ) ≥ n. Note:
χ(Gc ) may be considered to be the smallest number of cliques into which the vertices
of G can be partitioned.
2
Part B: (20 points each) Submit solutions to any two.
20 minutes.
Time: 1 hour and
Problem B1: Prove that, for positive integers n and |z| < 1,
∞ X
n+k−1 k
1
=
z .
(1 − z)n
k
k=0
Problem B2: Let G = (V, E) be a simple graph with vertex connectivity κ(G).
a) Prove that if κ(G) ≥ |V2 | , then G is hamiltonian.
b) For each positive integer r, give an example of a non-hamiltonian graph of
order |V | = 2r + 1 such that κ(G) = r.
Problem B3: A maximal planar graph is a simple planar graph having the property
that adding any edge between non-adjacent vertices destroys planarity.
a) Prove that if a maximal planar graph has no vertices of degree larger than
6, then
3n3 + 2n4 + n5 = 12,
where nd denotes the number of vertices of degree d for d = 3, 4, 5.
b) Does there exist a maximal planar graph having vertices of degree 3 and 5
only, with the same number of each?
c) Does there exist a maximal planar graph having vertices of degree 4 and 5
only, with the same number of each?
3
Combinatorics Qualifying Exam
August, 2008
This examination consists of two parts, A and B. Part A contains six problems of which you
must select four to do. Part B contains three problems of which you must select two to do. Each
problem in part A is worth 15 points and each problem in part B is worth 20 points. Only hand-in
your solutions to four problems from part A and two from part B. Please do not turn-in more solutions
since only the first four solutions from part A will be graded and only the first two solutions from
part B will be graded.
Begin each problem on a new sheet of paper and be sure to label each page of your work with
the problem number and your name.
In each question, if you appeal to a theorem within your solution, you must carefully state the
entire theorem. All graphs, unless otherwise stated, should be understood to be finite and simple.
1
Part A: (15 points each) Do any four.
Time: 2 hours.
Problem A1: A tournament is a complete graph in which every edges has been given an orientation.
Prove that every tournament has a directed Hamiltonian path.
Problem A2: Solve the recurrence
an = 5an−1 − 6an−2
( for n ≥ 2),
with initial conditions a0 = 1 and a1 = 1.
Problem A3: Suppose that G is a connected planar graph that can be drawn in the plane so that
all faces have an even number edges on their boundary. Prove that the vertices of G can be properly
2-colored.
Problem A4: All points of the plane that have integer coordinates are colored so that each such
point receives one of the three colors: red, blue or green. Prove that there must be a rectangle whose
four corner vertices are all of the same color.
Problem A5: Prove that if every chain and every antichain of a poset P is finite, then P is finite.
Problem A6: Let G be a graph in which any two odd cycles intersect.
a) Prove that G is 5-colorable.
b) Give an example to show that 4 colors do not suffice.
2
Part B: (20 points each) Do any two.
Time: 1 hour and 20 minutes.
Problem B1:
a) Find, with a proof, the number of edges in the extremal graph on 6 vertices without K4 as
a subgraph.
b) Find, with a proof, the number of edges in the extremal graph on 6 vertices without C4 as
a subgraph.
Problem B2: Prove the given identity:
a)
¶
n µ ¶µ
X
a
b
i=0
i
n−i
=
µ
¶
a+b
n
b)
µ ¶
n
X
n
k
k=1
k
= n2n−1 .
Problem B3: Prove or disprove: If G is a connected, simple graph that does not contain P4 or C3
as an induced subgraph, then G is a complete bipartite graph.
3
Preliminary Exam
COMBINATORICS
May, 2006
This examination consists of two parts, A and B. Part A consists of five problems
and part B consists of three problems. Each problem in part A is worth 15 points
and each problem in part B is worth 20 points. You have to solve any four
problems out of part A and any two problems out of part B. Begin each problem on
a new sheet of paper, and only write on one side of the paper. Only hand in those
selected six problems. You have 3 hours and 30 minutes to complete the exam.
PART A (15 points each) Do any four.
Problem A1.
Let G be a simple graph with n vertices, n ≥ 3.
(a) Determine, with a proof, all graphs G having the property that G – e is a tree
for every edge e ∈ E (G) . Give an example of such a graph of order n = 5.
(a) Characterize those graphs G for which G – e is a tree for some edge
e ∈ E (G) . Give an example of such a graph of order n = 5 different than an
example
€ in part (a).
€ Problem A2.
For a graph G, let α (G) denote the maximum size of an independent set of vertices
in G. Suppose that G is a bipartite graph of order 2m.
Prove: α (G) = m if and only if G has a perfect matching.
€
€Problem A3.
A diameter, diam(G), of a graph G is the length of the longest path in G.
χ (G) is the chromatic number of G.
(a) Prove that χ (G) ≤ diam(G) + 1.
(a) Give an example of a graph G for which χ (G) = diam(G) + 1.
(a) Show that the difference between the numbers diam(G) + 1 and χ (G) can be
€
arbitrarily
large.
€
€
€
Problem A4.
A caterpillar is a tree having the property that after deleting all leaves (vertices of
degree 1) from it, the remaining graph is a path. A diameter of a tree, diam(T), is
the length of the longest path.
Show that if T is a caterpillar of order n with diam(T) = k (k < n), then its
independence number α (G) ≥ n − k + 1.
€
€
Problem A5.
Let an denote the number of n-digit sequences in which each digit is 0, 1, or –1,
with no two consecutive 1s or two consecutive –1s allowed.
Prove that an satisfies the recurrence relation
an = 2an−1 + an−2 , n ≥ 3, and find a formula for an .
€
€
€
PART B (20 points each) Do any two.
€
Problem B1.
An n × n × n cube consists of n 3 unit cubes stacked into a rectangular pile
having width, length, and height n. Two units cubes are adjacent if they
share a 2-dimensional face.
Determine with a proof
€ all values of n, n ≥ 2, for which it is possible to list
all unit cubes in such a way that all three conditions are satisfied:
(1) no cube is repeated;
(2) every two consecutive cubes in the listing are adjacent;
(3) the last cube and the first cube in the listing are adjacent.
Problem B2.
(a) Find a formula for the number of solutions of x1 + x 2 + K + x k < n ,
where n, xi are positive integers and k is fixed.
(b) Find a formula for the number of solutions of x1 + x 2 + K + x k = n ,
where x i = ±1 , n and k are fixed positive
€ integers.
€
Problem
€ B3.
Five differently colored dice are thrown simultaneously and the numbers of
dots on them are added.
(a) Use the ordinary generating function to find the number of outcomes
with the sum of dots equal to 22.
(b) Use the ordinary generating function to find the number of outcomes
with the sum of dots equal to 22 and even number of dots on each die.
Combinatorics Qualifying Exam
May 2005
This examination consists of two parts, A and B. Part A contains six problems of which you
must select four to do. Part B contains three problems of which you must select two to do. Each
problem in part A is worth 15 points and each problem in part B is worth 20 points. Only hand-in
your solutions to four problems from part A and two from part B. Begin each problem on a new sheet
of paper and be sure to label each page of your work with the problem number and your name.
In each question, if you appeal to a theorem within your solution, you must carefully state that
theorem. All graphs, unless otherwise stated, should be understood to be finite and simple.
1
Part A: (15 points each) Do any four.
Time: 2 hours.
Problem A1: Let T be a finite tree in which there are no vertices of degree two. Recall that a vertex
of degree one in a tree is called a leaf.
a) Prove that at least half of the vertices of T are leaves.
b) Prove that if T is sufficiently large, then it must contain a set S of 100 leaves with the
property that all distances (in T ) between elements of S are equal modulo 3.
Problem A2: Prove that there are no 4-regular bipartite planar graphs.
Problem A3: A set of integers A is fat if each of its elements is ≥ |A|. For example, the empty set
and {5, 7, 91} are fat, but {3, 5, 10, 14} is not. Let fn denote the number of fat subsets of {1, . . . , n}.
a) Find a recurrence relation for fn .
b) Find an explicit formula for fn . Justify your answer.
Problem A4: Recall that a total order is a partial order in which all pairs are comparable. Suppose
that P1 and P2 are two total orders on a set of n2 + 1 elements. Show that there is a subset of size
n + 1 on which P1 and P2 totally agree or totally disagree.
Problem A5: Use exponential generating functions to find the number of ways to distribute n
distinguishable balls to five different boxes with a positive even number of balls distributed to box 5.
Problem A6: Find the minimum number of edges whose removal from K6 leaves a planar graph.
Justify your claim.
2
Part B: (20 points each) Do any two.
Time: 1 hour and 20 minutes.
Problem B1:
a) Find the number of ways of giving 3n different toys to Maddy, Jimmy, and Tommy so that
Maddy and Jimmy together get 2n toys.
b) Find the number of solutions in nonnegative integers of the equation
a + b + c + d + e + f = 20
in which no variable is greater than 8.
Problem B2: Prove the given statement or provide a counterexample. Justify your claims.
a) Every connected graph with at least three vertices has at least two vertices whose removal
leaves a connected graph.
b) If a graph is cubic and has a hamiltonian path, then its edge-chromatic number is three.
c) If G (V, E) is a finite simple graph, then {(e, f) ∈ E × E: e and f are equal or lie on a common cycle}
is an equivalence relation on the edges of G.
Problem B3: Consider a finite collection of lines drawn in the plane so that no two lines are parallel
and no three lines share a point. Consider their points of intersection as the vertices of a graph and
the segments between neighboring intersection points as edges of our graph. Prove that this resulting
planar graph is 3-colorable.
3
Combinatorics Qualifying Exam
October, 2005
This examination consists of two parts, A and B. Part A contains six problems of which you
must select four to do. Part B contains three problems of which you must select two to do. Each
problem in part A is worth 15 points and each problem in part B is worth 20 points. Only hand-in
your solutions to four problems from part A and two from part B. Begin each problem on a new sheet
of paper and be sure to label each page of your work with the problem number and your name. You
have two hours to complete part A and one hour and 20 minutes to complete part B. There will be
ten-minute break between parts A and B.
In each question, if you appeal to a theorem within your solution, you must carefully state that
theorem. All graphs, unless otherwise stated, should be understood to be finite and simple.
1
Part A: (15 points each) Do any four.
Time: 2 hours.
Problem A1: An orientation of a graph G is a digraph D obtained by inserting an arrow on each
edge of G. Prove that a graph G always has an orientation D such that
−
| deg+
D (v) − degD (v)| ≤ 1,
for every vertex v of G.
Problem A2: Acme Airlines has n different routes (numbered 1 through n). A schedule set in
advance gives the starting time si and the finish time fi for each route i. Let tij be the time required
to move an airplane from destination route i to the origin or route j. This partially orders the routes:
place (i, j) in the partial order P if and only if fi + tij < sj ; that is, routes i and j are comparable iff
a single plane can run both routes.
a) State Dilworth’s Theorem.
b) What is the minimum number of planes needed to fly Acme’s routes?
Problem A3: Consider the alphabet X = {a, b, c}. Let wn denote the number words (sequences) of
length n over the alphabet X in which the letter b appears an even number of times.
a) Find the exponential generating function for wn .
b) Find a compact formula for wn .
Problem A4: The integer 3 can be expressed as an ordered sum of positive integers in four ways,
namely, 3, 2 + 1, 1 + 2, and 1 + 1 + 1. Prove that any positive integer n can be expressed as an ordered
sum of positive integers in 2n−1 ways.
Problem A5: A subset of the set {1, . . . , n} is alternating if its elements, when arranged in increasing
order, follow the pattern: odd, even, odd, etc. For example, {3}, {1, 2, 5}, and {3, 4} are alternating subsets of {1, 2, 3, 4, 5}, whereas {2, 3, 4, 5} and {1, 3, 4} are not. The empty set is considered
alternating. Let an denote the number of alternating subsets of {1, . . . , n}.
a) Find a recurrence for an .
b) Solve the recurrence in part (a) and find a formula for an .
Problem A6: The n-cube Qn (for n ≥ 1) is the graph whose vertices are the binary words of length n
and two vertices are joined by an edge if and only if their corresponding binary words differ in exactly
one coordinate. Show that Qn is planar if and only if n ≤ 3.
2
Part B: (20 points each) Do any two.
Time: 1 hour and 20 minutes.
Problem B1: Evaluate the given sum. Justify your answer:
a)
n
X
2r
r=0
n
r
b)
1
n
1
+2
n
2
+ ··· + n
n
.
n
Problem B2:
a) Prove that if G is a 3-regular simple graph then the vertex-connectivity of G is equal to the
edge-connectivity of G.
b) Prove that if G is a simple graph on n vertices with minimum degree δ ≥
n+k−2
,
2
then G is
k-connected.
Problem B3: An r × s Latin rectangle based on 1, . . . , n is an r × s matrix such that each entry is
one of the integers 1, . . . , n and each integer occurs in each row and column at most once. Prove that
every r × n Latin rectangle based on 1, . . . , n can be extended (by adding rows) to an n × n Latin
square. (Hint: Do induction on r. Create an appropriate bipartite graph and show the existence of a
perfect matching in it to extend the Latin rectangle).
3
Combinatorics Qualifying Exam
October, 2004
This examination consists of two parts, A and B. Part A contains five problems of which you
must select four to do. Part B contains three problems of which you must select two to do. Each
problem in part A is worth 15 points and each problem in part B is worth 20 points. Only hand-in
your solutions to four problems from part A and two from part B. Begin each problem on a new sheet
of paper and be sure to label each page of your work with the problem number and your name. You
have two hours to complete part A and one hour and 20 minutes to complete part B. There will be
ten-minute break between parts A and B.
In each question, if you appeal to a theorem within your solution, you must carefully state that
theorem. All graphs, unless otherwise stated, should be understood to be finite and simple.
1
Part A: (15 points each) Do any four.
Time: 2 hours.
Problem A1: A graph G is a chordal graph if every cycle C of G contains an edge joining two nonconsecutive vertices of C. A graph G(V, E) is an interval graph if there is an assigment f that assigns
an interval Iv of the real line to each vertex v ∈ V (G) such that uv ∈ E if and only if Iu ∩ Iv 6= ∅.
Prove that every interval graph is a chordal graph.
Problem A2: A tournament is an complete graph in which every edges has been given an orientation.
Prove that every tournament has a directed Hamiltonian path.
Problem A3: Solve the recurrence
an = 5an−1 − 6an−2
( for n ≥ 2),
with initial conditions a0 = 1 and a1 = 1.
Problem A4: Suppose that G is a connected planar graph that can be drawn in the plane so that
all faces have an even number edges on their boundary. Prove that the vertices of G can be properly
2-colored.
Problem A5: Let hn denote the number of nonnegative integral solutions of the equation:
x1 + x2 + x3 + x4 = n.
a) Write the ordinary generating function for hn .
b) What is h25 ?
2
Part B: (20 points each) Do any two.
Time: 1 hour and 20 minutes.
Problem B1: Prove the given identity:
a)
¶
n µ ¶µ
X
a
b
i=0
i
n−i
=
µ
¶
a+b
n
b)
µ ¶
n
X
n
k
k=1
k
= n2n−1 .
Problem B2:
a) State the definition of what it means for a graph to be a perfect graph.
b) A graph G(V, E) is a comparability graph if there is a partial order P on V so that uv ∈ E if
and only if u and v are comparable in P . Prove that every comparability graph is perfect.
Problem B3: Use inclusion-exclusion to find a formula for the number of 1-factors in the graph
obtained from Kn,n by removing the edges of a perfect matching.
3
Combinatorics Qualifying Examination
May 28, 2003
This examination consists of two parts, A and B. Part A consists of five problems
and Part B consists of three problems. You are to do any four problems from Part
A and any two problems from Part B. Each problem from Part A is valued at 15
points, and each problem in Part B is worth 20 points. Only hand in four problems
from Part A and two problems from Part B. Begin each problem on a new sheet of
paper, and only write on one side of the paper. You have two hours to complete Part
A of the exam. When you are ready to hand in your exam, assemble the problems
in numerical order, write your name on the front page, and initial the other pages.
There will be a ten minute break before Part B. You have one hour and 20 minutes
to complete Part B of the exam.
1
Part A (15 points each) Do any four.
A1. Let G be a connected simple graph whose line graph L(G) is cubic.
(a) Prove that for every edge e = uv of G, degG u + degG v = 5.
(b) Prove that the graph G is bipartite.
A2. The Cartesian product of graphs G and H, written G × H, is the graph with
vertex set V (G) × V (H), and two vertices (u, v) and (x, y) of G × H are adjacent if
and only if either (1) u = x and vy ∈ E(H), or (2) v = y and ux ∈ E(G).
The n-cube Qn is defined by Q1 = K2 and Qn = Qn−1 × K2 for n ≥ 2. Let G be a
graph with chromatic number χ(G) = k ≥ 2.
(a) Let C5 denote the 5-cycle. Draw a plane embedding of the graph C5 × K2 .
(b) Prove that χ(G × K2 ) = k.
(c) Prove that χ(G × Qn ) = k for each n ≥ 1.
A3. A graph is hamiltonian if it contains a spanning cycle (a cycle through every
vertex). A hamiltonian path is a spanning path (a path through every vertex). Prove
or disprove the following:
(a) Every cubic hamiltonian graph has edge-chromatic number 3.
(b) There exists a cubic eulerian graph with edge-chromatic number 3.
(c) Every cubic graph possessing a hamiltonian path has edge-chromatic number 3.
A4. Let X = {a, b, c}. Find the number N (n) of words (sequences) of length n
in which the letters are taken from X and the letter a appears an even number of
times. Use two different counting techniques:
(a) Exponential generating functions.
(b) Justify that N (n) satisfies the following recurrence relation: N (n + 1) = N (n) +
3n . Prove the compact formula for N (n) by induction.
A5. The crossing number of a graph G is the minimum number of crossings in a
drawing of G in the plane. Let G be the complete bipartite graph K4,3 .
(a) Prove that G is not a planar graph.
(b) Prove that the crossing number of G is not 1.
Hint: Suppose there is a drawing of G in the plane with one crossing v. Consider
the new (plane) graph H with one extra vertex v. Use Euler’s formula to find
the number of regions of H. What are the degrees of the faces of H? Obtain a
contradiction.
(c) Prove that the crossing number of G is at most 2.
2
Part B (20 points each) Do any two.
B1. If G is a simple graph with the vertex set V = {v1 , v2 , . . . , vn }, then its
adjacency matrix is the n × n matrix A = (aij ), where
aij = 1 if vi vj is an edge of G, and aij = 0 otherwise.
Let G be a simple (5, q) graph, with an adjacency matrix A. Suppose that

2
1


A2 = 1

2
2
1
4
3
2
2
1
3
4
2
2
2
2
2
3
2


2
2


2 ,

2
3
2
7


A3 = 7

4
4
7
8
9
9
9
7
9
8
9
9
4
9
9
6
7

4
9


9 .

7
6
(a) How many (non-identical) 3-cycles does G contain?
(b) Determine q, the number of edges of G.
(c) Determine diam G, the diameter of G.
(d) Determine rad G, the radius of G.
(e) Draw the graph of G.
B2. Consider the poset (P, ≤), where
P = {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72}
and for a, b ∈ P , a ≤ b if and only if a divides b.
(a) Represent P graphically by its Hasse disgram.
(b) Find all maximal and all minimal elements of (P, ≤). Give an example of a
maximum chain and an example of a maximum antichain in (P, ≤).
(c) State Dilworth’s theorem and illustrate it using the above poset P as an example.
(d) The cardinality of poset P is 16. Show that every poset of cardinality 16 must
contain either a chain of cardinality 6 or an antichain of cardinality 4.
3
B3. Consider the graph G with 34 vertices and 54 edges presented below.
v
v
v
vz
v
v
v
v
v
v
v
v
vy
v
v
v
v
v
v
v
v
v
x
b v
v
v
v
v
v
a v
v
v
v
v
v
(a) What is the maximum number of pairwise vertex internally-disjoint b, y-paths
in G?
(b) What is the maximum number of pairwise edge-disjoint b, y-paths in G?
(c) Find the number of shortest (not necessarily disjoint) a, x-paths in G.
(d) Find the number of shortest (not necessarily disjoint) a, z-paths in G passing
through x.
(e) Find the number of shortest (not necessarily disjoint) a, z-paths in G.
(f) What is the length of a longest a, z-path in G?
Justify all your answers!
4
Combinatorics Qualifying Exam
October, 2003
This examination consists of two parts, A and B. Part A contains ¯ve problems of which you
must select four to do. Part B contains three problems of which you must select two to do. Each
problem in part A is worth 15 points and each problem in part B is worth 20 points. Only hand-in
your solutions to four problems from part A and two from part B. Begin each problem on a new sheet
of paper and be sure to label each page of your work with the problem number and your name. You
have two hours to complete part A and one hour and 20 minutes to complete part B. There will be
ten-minute break between parts A and B.
In each question, if you appeal to a theorem within your solution, you must carefully state that
theorem. All graphs, unless otherwise stated, should be understood to be ¯nite and simple.
1
Part A: (15 points each) Do any four.
Time: 2 hours.
Problem A1: Let G be a graph in which any two odd cycles intersect.
a) Prove that G is 5-colorable.
b) Give an example to show that 4 colors do not su±ce.
Problem A2: Find all 3-regular plane graphs in which all faces are triangles. Prove your list is
complete.
Problem A3: Let
S(n) = f(A; B) : ; µ A µ B µ f1; 2; : : : ; ngg:
a) Find a recurrence relation for an = jS(n)j.
b) Find a compact formula for jS(n)j. Justify your answer.
Problem A4: Prove that if every chain and every antichain of a poset P is ¯nite, then P is ¯nite.
Problem A5: Consider the ways to distribute n identical balls to ¯ve di®erent boxes with the ¯rst
four boxes receiving between 3 and 8 balls.
a) Write the ordinary generating function for the number of these distributions.
b) In how many ways can 25 balls be distributed in this way?
2
Part B: (20 points each) Do any two.
Time: 1 hour and 20 minutes.
Problem B1: Prove the given identity:
a)
n
X
k=0
µ ¶
1
n
k+1 k
=
2n+1 ¡ 1
n+1
b)
n µ ¶2
X
n
k
k=0
=
µ ¶
2n
n
Problem B2:
a) State the de¯nition of what it means for a graph to be an interval graph.
b) State the de¯nition of what it means for a graph to be a perfect graph.
c) Prove directly that every interval graph is perfect.
Problem B3: Use inclusion-exclusion to ¯nd a formula for the number of 1-factors in the graph
obtained from Kn;n by removing the edges of a perfect matching.
3