Combinatorics Qualifying Exam August, 2016Combinatorics Qualifying Exam August, 2016 This examination consists of two parts, Combinatorics and Graph Theory. Each part contains ve problems

Combinatorics Qualifying ExamAugust, 2016

This examination consists of two parts, Combinatorics and Graph Theory. Each partcontains five problems of which you must select three to do. Each problem is worth 20points. Only hand in your solutions to three problems from each part. Please do not turnin 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 workwith the problem number and your name.

In each question, if you appeal to a theorem within your solution, you must carefullystate the entire theorem. All graphs, unless otherwise stated, should be understood to befinite and simple.

Arithmetic expressions need not be completely reduced unless otherwise stated, andmay be left in terms of sums, differences, products, quotients, exponentials, factorials, andbinomial 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∑k=0

(n

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 nelements of [n2 + 1] such that for all i, j ∈ A, π(i) < π(j) if and only if σ(i) < σ(j), or thatthere is a set B of n elements of [n2 + 1] such that for all i, j ∈ B, π(i) < π(j) if and only ifσi > 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 samesum.

2

Graph Theory. Do any three.

Problem B1: Prove that every graph G on 2n vertices with minimum degree δ(G) ≥ ncontains a perfect matching.

Problem B2: Prove that if G is a graph on n vertices with complement G, it is the casethat χ(G) + χ(G) ≥ 2

√n.

Problem B3: Let G be a cubic graph with edge-chromatic number 3 such that the partitionof 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). Provethat a tree T contains a vertex v from which every other vertex is at a distance of at mostdd(T )

2e.

Problem B5: Prove that if G is a graph with at least 11 vertices, then either G or itscomplement 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. Eachproblem is worth 20 points. Only hand in your solutions to three problemsfrom each part. Please do not turn in more solutions since only the first threesolutions from each part will be graded.

Begin each problem on a new sheet of paper and be sure to label eachpage of your work with the problem number and your name.

In each question, if you appeal to a theorem within your solution, youmust 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 sufficientlylarge integer n,

d∑

m=0

(

n

m

)(

n−m

d−m

)(

n− d

d−m

)

(d−m)!pd−m < 2

(

n

d

)(

n− d

d

)

d!pd

2. A non-crossing pairing of the set N = {1, . . . , 2n} is a partition of Ninto 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 that S(n, k) is the Stirling number of the second kind, provethat

∑

n

m=1

(

n

m

)

S(n − m, k) = (k + 1)S(n, k + 1) for all nonnegativeintegers n and k.

4. Prove that for every set of n integers there is a nonempty subset thatsums to a multiple of n.

5. Let an be the number of strings (or words) of length n consisting of theletters A, B, C, and D which contain at least one A and at least oneD. 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 generalizedcycle 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 monochro-matic spanning tree.

3. Suppose G has at least 5 vertices and that every induced subgraph on 3vertices has the same number of edges. Show that G is either completeor 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 asubset Y ⊆ B such that |N(Y )| < |Y | − b. Show that G does not havea 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


This examination consists of two parts, Combinatorics and Graph Theory. Each partcontains five problems of which you must select three to do. Each problem is worth 20points. Only hand in your solutions to three problems from each part. Please do not turnin more solutions since only the first three solutions from each part will be graded.



Arithmetic expressions need not be completely reduced unless otherwise stated, andmay be left in terms of sums, differences, products, quotients, exponentials, factorials, andbinomial coefficients.

1

Combinatorics. Do any three.

Problem A1: Prove that for nonnegative integer n and integer r ≥ 2:

n∑i=0

(n

i

)2

ri =n∑j=0

(n

j

)(2n− jn

)(r − 1)j

Problem A2: Prove that for any positive integer n there are the same number of partitionsof n into any number of parts none of which are divisible by 3 as there are partitions of ninto any number of parts such that the same number does not appear more than twice inthe partition.

Problem A3: Let k be an arbitrary positive integer. Prove that k has a positive multiplein 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 checkerboardwith 3 colors, if each color must be used at least once and two boards are considered to beidentical 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 ofsufficiently large dimension.

(b) Let h(P ) denote the least positive integer n such that P can be embedded into ann-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 injectivefunction f such that x ≤ y if and only if f(x) ≤ f(y).

2


Problem B1: Prove that a graph with n vertices and independence number α contains apath on at least dn

αe 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 possibleto label the leaves u1, u2, . . . , uk and v1, v2, . . . , vk such that all k of the unique paths fromui to vi have a vertex in common.

Problem B3: A graph is greedy-set colored by the following method: we repeatedly selectthe largest independent set of uncolored vertices (choosing an arbitrary set if there areseveral) 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 mostχ(G) + k colors.

Problem B4: Prove that there is a coloring of the edges of a K6,6 with two colors suchthat 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 acycle passing through all of them.

3


This examination consists of two parts, Combinatorics and Graph Theory. Each partcontains five problems of which you must select three to do. Each problem is worth 20points. Only hand in your solutions to three problems from each part. Please do not turnin more solutions since only the first three solutions from each part will be graded



1

Combinatorics. Do any three.

Problem A1: Let S = {1, 2, . . . , n}. A subset A of S is called a triple if |A| = 3. Thetriples A and B are independent if |A ∩ B| ≤ 1. Prove that if F = {A1, A2, . . . , Am} is acollection 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× 2and 2×2 rectangles. The chessboard is considered to be unrotatable, but the dominoes maybe 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 becolored with n different colors, if each color must be used at least once and if two coloringsare considered to be equivalent if one is a rotation of the other. (Hint: there are 60 ways torotate of a dodecahedron onto itself,

Problem A4: For a given partition q of the integer n, let t(n, q) be the number of copiesof the number 2 appearing in q, and let u(n, q) be the number of parts which appear exactlyonce 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 respectivelyhave 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∑n

k=0 k2(nk

).

2


Problem B1: Prove that if G contains at least one cycle and has girth at least 5, then thecomplement of G is Hamiltonian.

Problem B2: Determine, with proof, a formula for ex(n, P4), where P4 is the path on fourvertices, 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 vertex-disjoint union of α(G) (not necessarily induced) subgraphs of G, each of which is a cycle, anisolated 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 thefollowing property: for all disjoint A,B ⊆ V (G) with |A| = |B| = 2n, there are verticesa ∈ A and b ∈ B such that ab ∈ E(G), and there are also vertices a′ ∈ A and b′ ∈ B suchthat a′b′ 6∈ E(G).

3

Combinatorics Qualifying ExamJanuary 7, 2011

This examination consists of two parts, A and B. Part A contains six problemsof which you must select four to do. Part B contains three problems of which youmust select two to do. Each problem in part A is worth 15 points and each problemin part B is worth 20 points. Only hand-in your solutions to four problems from partA and two from part B. Please do not turn-in more solutions since only the first foursolutions from part A will be graded and only the first two solutions from part B willbe graded.

Begin each problem on a new sheet of paper and be sure to label each page ofyour work with the problem number and your name.

In each question, if you appeal to a theorem within your solution, you mustcarefully state the entire theorem. All graphs, unless otherwise stated, should beunderstood 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 ofthe 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 andno four of them are pairwise indivisible.

b) Prove that a set of 10 numbers contains either four pairwise divisible num-bers or four pairwise indivisible numbers.

Problem A2: Find the number of ways of arranging the 26 letters of the alphabetso that no one of the sequences ABC, PQRS, 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, whichcover two squares. Let bn be the number of possible configurations of objects so thatevery 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 graphKn,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. Showthat 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 verticesof G can be partitioned.

2

Part B: (20 points each) Submit solutions to any two. Time: 1 hour and20 minutes.

Problem B1: Prove that, for positive integers n and |z| < 1,

1

(1− z)n=

∞∑k=0

(n+ k − 1

k

)zk.

Problem B2: Let G = (V,E) be a simple graph with vertex connectivity κ(G).

a) Prove that if κ(G) ≥ |V |2 , then G is hamiltonian.

b) For each positive integer r, give an example of a non-hamiltonian graph oforder |V | = 2r + 1 such that κ(G) = r.

Problem B3: A maximal planar graph is a simple planar graph having the propertythat adding any edge between non-adjacent vertices destroys planarity.

a) Prove that if a maximal planar graph has no vertices of degree larger than6, 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


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)

nXi=0

µa

i

¶µb

n− i

¶=

µa+ b

n

¶b)

nXk=1

k

µn

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 ExamCOMBINATORICS

May, 2006

This examination consists of two parts, A and B. Part A consists of five problemsand part B consists of three problems. Each problem in part A is worth 15 pointsand each problem in part B is worth 20 points. You have to solve any fourproblems out of part A and any two problems out of part B. Begin each problem ona new sheet of paper, and only write on one side of the paper. Only hand in thoseselected 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 treefor 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 anexample in part (a).

Problem A2.For a graph G, let

€

α G( ) denote the maximum size of an independent set of verticesin 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 bearbitrarily large.

Problem A4.A caterpillar is a tree having the property that after deleting all leaves (vertices ofdegree 1) from it, the remaining graph is a path. A diameter of a tree, diam(T), isthe length of the longest path.Show that if T is a caterpillar of order n with diam(T) = k (k < n), then itsindependence 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

€

n3 unit cubes stacked into a rectangular pilehaving width, length, and height n. Two units cubes are adjacent if theyshare a 2-dimensional face.Determine with a proof all values of n, n ≥ 2, for which it is possible to listall 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 + x2 +K+ xk < n,where n, xi are positive integers and k is fixed.

(b) Find a formula for the number of solutions of

€

x1 + x2 +K+ xk = n,where

€

xi = ±1, n and k are fixed positive integers.

Problem B3.Five differently colored dice are thrown simultaneously and the numbers ofdots on them are added.

(a) Use the ordinary generating function to find the number of outcomeswith 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.


May 2005




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


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 setand {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


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) IfG (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


October, 2005





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.



1


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)− deg−D(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 alternat-

ing 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


Problem B1: Evaluate the given sum. Justify your answer:

a)

n∑r=0

2r

(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−22

, 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


October, 2004

This examination consists of two parts, A and B. Part A contains five problems of which you




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.



1


Problem A1: A graph G is a chordal graph if every cycle C of G contains an edge joining two non-

consecutive 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


Problem B1: Prove the given identity:

a)

nXi=0

µa

i

¶µb

n− i

¶=

µa+ b

n

¶b)

nXk=1

k

µn

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.

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Combinatorics Qualifying Examination

May 28, 2003

This examination consists of two parts, A and B. Part A consists of five problemsand Part B consists of three problems. You are to do any four problems from PartA and any two problems from Part B. Each problem from Part A is valued at 15points, and each problem in Part B is worth 20 points. Only hand in four problemsfrom Part A and two problems from Part B. Begin each problem on a new sheet ofpaper, and only write on one side of the paper. You have two hours to complete PartA of the exam. When you are ready to hand in your exam, assemble the problemsin 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 minutesto complete Part B of the exam.

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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 withvertex set V (G)× V (H), and two vertices (u, v) and (x, y) of G×H are adjacent ifand 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 agraph 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 everyvertex). A hamiltonian path is a spanning path (a path through every vertex). Proveor 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 nin which the letters are taken from X and the letter a appears an even number oftimes. 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 adrawing 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. Considerthe new (plane) graph H with one extra vertex v. Use Euler’s formula to findthe number of regions of H. What are the degrees of the faces of H? Obtain acontradiction.(c) Prove that the crossing number of G is at most 2.

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Part B (20 points each) Do any two.

B1. If G is a simple graph with the vertex set V = {v1, v2, . . . , vn}, then itsadjacency matrix is the n× n matrix A = (aij), where

aij = 1 if vivj is an edge of G, and aij = 0 otherwise.

Let G be a simple (5, q) graph, with an adjacency matrix A. Suppose that

A2 =

2 1 1 2 21 4 3 2 21 3 4 2 22 2 2 3 22 2 2 2 3

, A3 =

2 7 7 4 47 8 9 9 97 9 8 9 94 9 9 6 74 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 amaximum 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 mustcontain either a chain of cardinality 6 or an antichain of cardinality 4.

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B3. Consider the graph G with 34 vertices and 54 edges presented below.

vvvv

vvvv

vvvv

vvvv

vvvv

vvvvv

vvv

vvv

vvv

a

b

x

y

z

(a) What is the maximum number of pairwise vertex internally-disjoint b, y-pathsin 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 passingthrough 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!

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Combinatorics Qualifying ExamOctober, 2003

This examination consists of two parts, A and B. Part A contains ¯ve problems of which youmust select four to do. Part B contains three problems of which you must select two to do. Eachproblem in part A is worth 15 points and each problem in part B is worth 20 points. Only hand-inyour solutions to four problems from part A and two from part B. Begin each problem on a new sheetof paper and be sure to label each page of your work with the problem number and your name. Youhave two hours to complete part A and one hour and 20 minutes to complete part B. There will beten-minute break between parts A and B.

In each question, if you appeal to a theorem within your solution, you must carefully state thattheorem. All graphs, unless otherwise stated, should be understood to be ¯nite and simple.

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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 iscomplete.

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 ¯rstfour 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?

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Problem B1: Prove the given identity:a)

nX

k=0

1k + 1

µnk

¶=

2n+1 ¡ 1n + 1

b)

nX

k=0

µnk

¶2

=µ2nn

¶

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 graphobtained from Kn,n by removing the edges of a perfect matching.

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Combinatorics Qualifying Exam August, 2016Combinatorics Qualifying Exam August, 2016 This examination consists of two parts, Combinatorics and Graph Theory. Each part contains ve problems