Combinatorics Qualifying Exam August, 2016

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1 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

2 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 S(k, m) = S(n + 1, m + 1) k where S(, ) is the Stirling number of the second kind. Problem A3: Let a n 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. a 0 = 0, a 1 = 1, and a 2 = 6). Find a closed formula for a n. Problem A4: Let π and σ be elements of S n Prove that there is either a set A of n elements of [n 2 + 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 [n 2 + 1] such that for all i, j B, π(i) < π(j) if and only if σ i > σ j 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

3 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) = max u,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 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

4 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 fromeach 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

5 Combinatorics Do any three. 1. Let d > 0 be an integer, and 0 < p < 1. Show that for a sufficiently large integer n, d ( )( )( ) ( )( ) n n m n d n n d (d m)!p d m < 2 d!p d m d m d m d d 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 that S(n, k) is the Stirling number of the second kind, prove that n n m=1( m) S(n m,k) = (k + 1)S(n,k + 1) for all nonnegative 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 a n 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 a n. 2

6 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 K n 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 e 1,e 2,e 3 such that G {e 1,e 2,e 3 } has no cycles. Prove that G is planar. 3

7 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

8 Combinatorics. Do any three. Problem A1: Prove that for nonnegative integer n and integer r 2: n i=0 ( ) 2 n r i = i n j=0 ( n j )( 2n j n ) (r 1) j 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 A k denote the k-element antichain. Find h(a 12 ). 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

9 Graph Theory. Do any three. Problem B1: Prove that a graph with n vertices and independence number α contains a path on at least n vertices (note that a single vertex may be considered to be a degenerate α 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 u 1, u 2,..., u k and v 1, v 2,..., v k such that all k of the unique paths from u i to v i 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 K 6,6 with two colors such that there is not a monochromatic K 3,3. 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

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13 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

14 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 = {A 1, A 2,..., A m } 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, q) be the number of parts which appear exactly once in q. Let a n = q t(n, q) and b n = q u(n, q). For example, 5 may be partitioned into 5, 4 + 1, 3 + 2, , , , and These respectively have t(5, q) values of 0, 0, 1, 0, 2, 1, and 0, for a total of a 5 = 4, while the values of u(5, q) are respectively 1, 2, 2, 1, 1, 1, and 0, so b 5 = 8. Prove that a n = b n 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 a n and b n 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( n k). 2

15 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, P 4 ), where P 4 is the path on four vertices, and ex represents the extremal number. Problem B3: Prove that every 4-connected graph contains a subdivision of K 4. 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 K 2, which covers every vertex of G. Problem B5: Prove that for all n 2 there exists a graph G on 2 n 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 a A and b B such that a b E(G). 3

16 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

17 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 b n be the number of possible configurations of objects so that every square is covered. a) Find a recurrence relation describing b n. b) Find a closed formula for b n. Problem A4: Prove that every spanning subgraph of the complete bipartite graph K n,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(g c ) n. Note: χ(g c ) may be considered to be the smallest number of cliques into which the vertices of G can be partitioned. 2

18 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, 1 (1 z) n = k=0 ( n + k 1 k ) z k. 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 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 3n 3 + 2n 4 + n 5 = 12, where n d 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

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22 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

23 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 a n =5a n 1 6a n 2 (forn 2), with initial conditions a 0 =1anda 1 =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 beagraphinwhichanytwooddcyclesintersect. a) Prove that G is 5-colorable. b) Give an example to show that 4 colors do not suffice. 2

24 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 K 4 as a subgraph. b) Find, with a proof, the number of edges in the extremal graph on 6 vertices without C 4 as a subgraph. Problem B2: Prove the given identity: a) b) nx i=0 µ µ a b = i n i µ a + b n nx µ n k = n2 n 1. k k=1 Problem B3: Prove or disprove: If G is a connected, simple graph that does not contain P 4 or C 3 as an induced subgraph, then G is a complete bipartite graph. 3

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31 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.

32 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 a n 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 a n satisfies the recurrence relation a n = 2a n 1 + a n 2, n 3, and find a formula for a n.

33 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 x 1 + x 2 +K+ x k < n, where n, x i are positive integers and k is fixed. (b) Find a formula for the number of solutions of x 1 + 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.

34 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

35 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) ProvethatatleasthalfoftheverticesofT 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: AsetofintegersA 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 f n denote the number of fat subsets of {1,...,n}. a) Find a recurrence relation for f n. b) Find an explicit formula for f n. Justify your answer. Problem A4: Recall that a total order is a partial order in which all pairs are comparable. Suppose that P 1 and P 2 are two total orders on a set of n elements. Show that there is a subset of size n +1onwhichP 1 and P 2 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 K 6 leaves a planar graph. Justify your claim. 2

36 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)isafinite 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

37 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

38 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 for every vertex v of G. deg + D (v) deg D (v) 1, Problem A2: Acme Airlines has n different routes (numbered 1 through n). A schedule set in advance gives the starting time s i and the finish time f i for each route i. Let t ij 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 f i + t ij < s j; 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 w n 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 w n. b) Find a compact formula for w n. 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 Prove that any positive integer n can be expressed as an ordered sum of positive integers in 2 n 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 a n denote the number of alternating subsets of {1,..., n}. a) Find a recurrence for a n. b) Solve the recurrence in part (a) and find a formula for a n. Problem A6: The n-cube Q n (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 Q n is planar if and only if n 3. 2

39 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 ( ) 2 r n r r=0 b) ( ) ( ) ( ) n n n n. 1 2 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

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42 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

43 Part A: (15 points each) Do any four. Time: 2 hours. Problem A1: AgraphG is a chordal graph if every cycle C of G contains an edge joining two nonconsecutive vertices of C. A graph G(V,E) isaninterval graph if there is an assigment f that assigns an interval I v ofthereallinetoeachvertexv V (G) such that uv E if and only if I u I v 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 a n =5a n 1 6a n 2 (forn 2), with initial conditions a 0 =1anda 1 =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 h n denote the number of nonnegative integral solutions of the equation: x 1 + x 2 + x 3 + x 4 = n. a) Write the ordinary generating function for h n. b) What is h 25? 2

44 Part B: (20 points each) Do any two. Time: 1 hour and 20 minutes. Problem B1: Prove the given identity: a) b) nx i=0 µ µ a b = i n i µ a + b n nx µ n k = n2 n 1. k k=1 Problem B2: a) State the definition of what it means for a graph to be a perfect graph. b) AgraphG(V,E) isacomparability 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 K n,n by removing the edges of a perfect matching. 3

45 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

46 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, deg G u + deg G 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 Q n is defined by Q 1 = K 2 and Q n = Q n 1 K 2 for n 2. Let G be a graph with chromatic number χ(g) = k 2. (a) Let C 5 denote the 5-cycle. Draw a plane embedding of the graph C 5 K 2. (b) Prove that χ(g K 2 ) = k. (c) Prove that χ(g Q n ) = 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) + 3 n. 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 K 4,3. (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

47 Part B (20 points each) Do any two. B1. If G is a simple graph with the vertex set V = {v 1, v 2,..., v n }, then its adjacency matrix is the n n matrix A = (a ij ), where a ij = 1 if v i v j is an edge of G, and a ij = 0 otherwise. Let G be a simple (5, q) graph, with an adjacency matrix A. Suppose that A 2 = , A 3 = (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

48 B3. Consider the graph G with 34 vertices and 54 edges presented below. z y x b a (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

49 Combinatorics Qualifying Exam October, 2003 This examination consists of two parts, A and B. Part A contains ve problems of which you must selectfour to do. Part Bcontains three problems of which youmust selecttwo to do. Each problem inparta is worth15 points andeach 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

50 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 listis complete. Problem A3: Let S(n) =f(a, B) :;µaµbµf1,2,..., ngg. a) Find a recurrence relation for a n =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

51 Part B: (20 points each) Do any two. Time: 1 hour and 20 minutes. Problem B1: Prove the givenidentity: a) b) nx k=0 µ 1 n = 2n+1 1 k+1 k n+1 nx k=0 µ 2 n = k 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 directlythateveryinterval graphis perfect. Problem B3: Use inclusion-exclusion to nd a formula for the number of 1-factors in the graph obtained from K n,n by removing the edges of a perfect matching. 3