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Michigan State University | MATH 481
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Michigan State University MATH 481
Journal of Combinatorial Theory, Series B 1750 journal of combinatorial theory, Series B 70, 2 44 (1997) article no. TB971750 The Four-Colour Theorem Neil Robertson* Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, -
Michigan State University MATH 481
MATH 481: HOMEWORK 5 (1) Find the coefficient of x6 in (1 + x + x2 )n . (2) Find the coefficient of x6 in (1 + x + x2 + x3 + . . . )n . (3) Suppose that a football team scored 100 points in a game, using only field goals (3 points) and touchdowns (7 -
Michigan State University MATH 481
MATH 481: HOMEWORK 5 SOLUTIONS (1) Find the coecient of x6 in (1 + x + x2 )n . Answer: We know that (1 + x + x2 )n = 0jk n k k j+k x j The pairs (j, k) such that j + k = 6 are (0, 6), (1, 5), (2, 4) and (3, 3). Therefore, the coecient of x6 is n -
Michigan State University MATH 481
MATH 481 SPRING 2009, MIDTERM EXAM 2 SOLUTIONS 1. How many ways are there to choose 4 cards from a quadruple deck? Hint: There are several ways to do this problem. Answer: Lets rephrase this problem in terms of bins: we are looking for the number of -
Michigan State University MATH 481
MATH 481: HOMEWORK 5 SOLUTIONS (1) Draw all rooted binary trees with 4 parent vertices, and use your drawings to explain the following equation: c4 = c3 c0 + c2 c1 + c1 c2 + c0 c3 Answer: Recall that ck is the number of rooted binary trees with k pa -
Michigan State University MATH 481
MATH 481: REVIEW 2 (1) How many ways are there to pick 5 cards from a triple deck? Answer: We are looking for the coecient of x7 in (1 + x + x2 + x3 )52 . One way to do this is to use the binomial theorem to get: (1 + x + x2 + x3 )52 = 0ijk52 52 k -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 17 VIVEK DHAND 1. Counting with symmetry A group G is a set equipped with a binary operation (written like multiplication) and a distinguished element e (called the identity) such that: (1) For any a, b G, we have ab G ( -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 27 VIVEK DHAND 1. Trees A tree is a connected graph with no cycles. A forest is a graph with no cycles, i.e. a graph whose connected components are trees. A vertex of degree one in a tree is called a leaf. Lemma. Let G be -
Michigan State University MATH 481
NOTES FOR MATH 481 MAPS VIVEK DHAND The number of elements in a nite set X will be denoted |X|. Given two sets X and Y we can form a new set X Y , called the disjoint union of X and Y . The elements of X Y are simply the elements of X and the elemen -
Michigan State University MATH 481
MATH 481 FALL 2008, MIDTERM EXAM 1 1. Prove the following identity: n n+1 + 2 2 Proof. n 2 = n! n(n 1)(n 2)! n(n 1) = = 2!(n 2)! 2(n 2)! 2 = n2 . Replacing n by n + 1, we get: n+1 2 Therefore, n n+1 + 2 2 = n2 n + n2 + n = n2 . 2 = n(n + 1) 2 -
Michigan State University MATH 481
MATH 481 REVIEW PROBLEMS FOR EXAM 3: SOLUTIONS (1) Let G be a k-regular graph of order n. What are the sizes of G and G? Answer: By denition, the degree of every vertex of G is k. By Theorem 1.1, the size of G is n n 1 1 kn deg(vi ) = k= 2 2 2 i=1 i -
Michigan State University MATH 481
MATH 481: REVIEW 1 SOLUTIONS (1) Use induction to prove that, for all n 1: n k 3 = T (n)2 k=1 where T (n) is the n-th triangular number. Proof. Recall that T (n) = If n = 1, then we have 13 = 1(1 + 1) 2 2 n(n + 1) 2 =1 Now assume the n-th stat -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 28 VIVEK DHAND Counting Labeled Trees A labeled tree is a tree whose vertices are ordered (for example, we could label them by the numbers 1, 2 . . . , n). Therefore, it makes sense to talk about the smallest or largest ve -
Michigan State University MATH 481
NOTES FOR MATH 481 CHROMATIC POLYNOMIALS VIVEK DHAND Let G be a graph of order n and size q. Let V = {v1 , . . . , vn } be the set of vertices of G and let E = {e1 , e2 , . . . , eq } be the set of edges of G. Given a subset S E, we dene N (S) to b -
Michigan State University MATH 481
NOTES FOR MATH 481 THE PRINCIPLE OF INCLUSION/EXCLUSION (PIE) VIVEK DHAND Let X be a set and A, B P (X). Recall that we dened the indicator function of a subset. This is a map IA : X {0, 1} dened by the formula: IA (x) = 1 if x A 0 if x A / Nex -
Michigan State University MATH 481
MATH 481: HOMEWORK 9 SOLUTIONS (1) If G is the graph in Figure 1.63, prove that (G) = 6 and (G) = 5. Proof. We can easily nd a K5 subgraph in G by taking the three vertices of degree 7 and any two of the degree 5 vertices. However, there is no K6 su -
Michigan State University MATH 481
NOTES FOR MATH 481 POLYAS FORMULA VIVEK DHAND In these notes we prove a localization formula due to Frobenius (also known as Burnsides lemma). We then use this to prove Plyas enumeration formula. Let G be o a nite group acting on a nite set X. For -
Michigan State University MATH 481
MATH 481: HOMEWORK 4 SOLUTIONS (1) Let a and b be positive integers. Prove that xab 1 = xa 1 b1 xak k=0 Proof. Recall that, for any positive integer b: zb 1 = z1 Now let z = xa : xab 1 = xa 1 b1 b1 zk k=0 xak k=0 (2) Find the coecient of x -
Michigan State University MATH 481
MATH 481: HOMEWORK 4 (1) Let a and b be positive integers. Prove that xab 1 = xa 1 (2) Find the coecient of x1000 in A(x) = 1 (1 x)(1 x2 )(1 x3 ) b1 xak k=0 (3) Suppose a basketball team scores 1000 points in a game using eld goals (2 or 3 po -
Michigan State University MATH 481
MATH 481: HOMEWORK 2 (1) Suppose that a certain password is a string of 8-12 characters consisting of uppercase letters, lowercase letters, and the digits 0-9. (a) How many possible passwords are there? (b) How many passwords are there if all the ch -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 11 VIVEK DHAND 1. Changing money Suppose we have a currency with n dierent kinds of coins, whose values are a1 , . . . , an . Theorem. The number of ways to make k cents in change is equal to the coecient of xk in the foll -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 26 VIVEK DHAND 1. Bipartite Graphs A graph G = (V, E) is bipartite if we can split up the vertices into two sets X and Y so that every edge has one endpoint in X and the other endpoint in Y . In other words, no two vertice -
Michigan State University MATH 481
NOTES FOR MATH 481 EULERIAN WALKS VIVEK DHAND 1. The Bridges of Konigsberg The problem of traversing each bridge exactly once can be translated into the problem of nding a walk in the following graph that traverses each edge exactly once: a b -
Michigan State University MATH 481
NOTES FOR MATH 481 GRAPHS VIVEK DHAND Graph theory has applications to a staggering number of rapidly growing elds, many of them at the cutting edge of science and technology. The theory of networks has really come into its own in the age of the int -
Michigan State University MATH 481
MATH 481: HOMEWORK 8 (1) Let G be a connected planar graph of order 24, and suppose it is regular of degree 3. How many regions are in a planar representation of G? (2) Prove that Eulers formula fails for disconnected graphs. (3) Let G be a planar g -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 1 VIVEK DHAND 1. Overview Discrete math is primarily about answering three types of questions: existence of a solution, counting the number of solutions, and optimization (finding the best solution). This course naturally -
Michigan State University MATH 481
NOTES FOR MATH 481 MATHEMATICAL INDUCTION VIVEK DHAND Mathematical induction simply says that if two expressions are equal to begin with and are given the same instructions, then they will always be equal. To make this precise, suppose we have two l -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 3 VIVEK DHAND 1. Counting For the sum and product rule, see Sec. 2.1 of the textbook. The number of possible words of length k using an alphabet of n letters is nk . This follows from the product rule: we have k sequential -
Michigan State University MATH 481
MATH 481: HOMEWORK 5 (1) Draw all rooted binary trees with 4 parent vertices, and use your drawings to explain the following equation: c4 = c3 c0 + c2 c1 + c1 c2 + c0 c3 (2) Write out the elements of the following groups using cycle notation: (a) C6 -
Michigan State University MATH 481
NOTES FOR MATH 481 OVERVIEW AND SET THEORY VIVEK DHAND 1. Overview Discrete math is primarily about answering three types of questions: existence of a solution, counting the number of solutions, and optimization (nding the best solution). This cours -
Michigan State University MATH 481
NOTES FOR MATH 481 BIPARTITE GRAPHS AND TREES VIVEK DHAND Bipartite Graphs Theorem. Let G be a graph of order n 2. G is bipartite if and only if it contains no odd cycles. Proof. Let G be a bipartite graph of order n 2. Let (v1 , . . . , vk , v1 ) -
Michigan State University MATH 481
MATH 481: HOMEWORK 1 (1) Let X = {a, b, c}. Write down the addition and multiplication tables for the corresponding ring of sets P (X). (2) Draw the divisor graph of 210. Compare with the subset graph of a set with four elements. (3) Show that if A -
Michigan State University MATH 481
MATH 481: HOMEWORK 2 (1) Find the number of possibilities for each kind of ve-card hand in poker: garbage, one pair, two pair, three of a kind, straight, ush, full house, four of a kind, straight ush. Explain your reasoning carefully, and make sure -
Michigan State University MATH 481
MATH 481: HOMEWORK 2 SOLUTIONS (1) Suppose that a certain password is a string of 8-12 characters consisting of uppercase letters, lowercase letters, and the digits 0-9. (a) How many possible passwords are there? (b) How many passwords are there if -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 5 VIVEK DHAND 1. Four types of collections Remember that we have four types of collections: ordered with repeats allowed, ordered with no repeats allowed, unordered with no repeats allowed, and unordered with repeats allow -
Michigan State University MATH 481
NOTES FOR MATH 481 FOURTH BASIC PROBLEM VIVEK DHAND The fourth basic problem is to count how many k-element multisets can be chosen from a set of n elements. Let X = {x1 , . . . , xn } be a set of n elements. A k-element multi-set chosen from X can -
Michigan State University MATH 481
MATH 481: REVIEW 1 (1) Use induction to prove that, for all n 1: n k 3 = T (n)2 k=1 where T (n) is the n-th triangular number. (2) How many possible outcomes are there if we roll a six-sided die 50 times? How many possible outcomes are there if w -
Michigan State University MATH 481
MATH 481: HOMEWORK 3 (1) Prove that: ab 2 = a 1 b a + 2 2 b 1 2 Give a combinatorial interpretation of this formula in terms of choosing 2 objects from the following set: S = {(i, j) | 1 i a, 1 j b} (2) Prove that: ab 3 = a 1 b a + 3 2 2 1 b 2 -
Michigan State University MATH 481
MATH 481 SPRING 2009, MIDTERM EXAM 1 SOLUTIONS 1. Use induction to prove that, for all n, m 0: n k=0 k m = n+1 m+1 Hint: assume m is fixed, and use induction on n. Proof. The base case is n = 0, so we have to show 0 1 = m m+1 for all m 0. Both -
Michigan State University MATH 481
NOTES FOR MATH 481 WALKS AND DIGRAPHS VIVEK DHAND 1. Review of matrix multiplication Suppose we have two lists of n numbers, say (a1 , . . . , an ) and (b1 , . . . , bn ). The dot product of the two lists is the number (a1 , . . . , an ) (b1 , . . -
Michigan State University MATH 481
MATH 481: HOMEWORK 8 SOLUTIONS (1) Let G be a graph of order n with vertices v1 , . . . , vn . The line graph L(G) is dened as follows: the vertices of L(G) are the edges of G. If e1 and e2 are two distinct edges of G, we draw an edge between them i -
Michigan State University MATH 481
MATH 481: HOMEWORK 6 (1) Find the order and size of Kn,m . (2) Prove that it is impossible for all the vertices of a graph to have dierent degrees. Hint: The proof is very short. (3) The friendship graph is dened as follows: the vertices are people, -
Michigan State University MATH 481
MATH 481 EXTRA CREDIT PROBLEMS Binary trees. Draw the following convex polyhedron: the vertices are labeled by rooted binary trees with five leaves. Draw an edge between two vertices if you can get from one tree to another by moving only one branch. -
Michigan State University MATH 481
MATH 481: HOMEWORK 9 (1) If G is the graph in Figure 1.63, prove that (G) = 6 and (G) = 5. (2) Let G be a graph of order n. Prove that (G)(G) n (Hint: Theorems 1.25 and 1.26) (3) Find the chromatic polynomials of the following graphs: (a) a tree T -
Michigan State University MATH 481
MATH 481 REVIEW PROBLEMS 1. How many possible outcomes are there if we roll a die fty times? How about if we roll fty dice at the same time? 2. Count the number of possibilities for each of the following situations: (a) The number of passwords of le -
Michigan State University MATH 481
MATH 481 SPRING 2009, MIDTERM EXAM 3 SOLUTIONS 1. (a) State the denition of a bipartite graph. Answer: Any one of the following statements is a correct answer to this question: G is bipartite if and only if we can split up the vertices into two sets -
Michigan State University MATH 481
MATH 481: DISCRETE MATHEMATICS I SPRING 2009 Course webpage: http:/www.math.msu.edu/dhand/MTH481SS09 Course Schedule: 1/12 Introduction 1/19 Holiday No Class 1/26 Binomial Theorem 2/2 Review Counting with repetition 2/16 Recurrences 2/23 Burnside\'s -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 2 VIVEK DHAND 1. Mathematical induction Mathematical induction simply says that if two expressions are equal to begin with and are given the same instructions, then they will always be equal. To make this precise, suppose -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 24 VIVEK DHAND 1. Graphs A graph G is a pair of finite sets (V, E) such that V is non-empty and E is a set of unordered pairs of elements of V . The elements of V are called vertices. The elements of E are called edges. Gi -
Michigan State University MATH 481
NOTES FOR MATH 481 LECTURE 25 VIVEK DHAND 1. Definitions Let G = (V, E) be a graph. A walk in G is a sequence of vertices (v0 , v1 , . . . , vk ) such that {vi , vi+1 } E. A path in G is a walk without repeated vertices. A cycle in G is a walk (v0 -
Michigan State University MATH 481
MATH 481: HOMEWORK 1 (1) Give an algebraic proof and a combinatorial proof of the following: T (n) + T (n - 1) = n2 For the algebraic proof, you may use what we proved in class: n(n + 1) T (n) = 2 (2) Use induction to prove that 1 2 + 2 3 + 3 4 + -
Michigan State University MATH 481
MATH 481: REVIEW 2 (1) How many ways are there to pick 5 cards from a triple deck? (2) How many ways are there to pick 100 coins from an infinite supply of pennies, nickels, dimes, and quarters? (3) Show that there are (m + 1)2 ways to make 10m cent -
Michigan State University MATH 481
MATH 481: HOMEWORK 7 (1) Find all integers n 1 such that Kn is bipartite. Find all integers n, m 1 such that Kn,m is a tree. (2) Let G be a graph. Prove that G is a tree if and only if, for any vertices u and v, there is exactly one path in G from -
Michigan State University MATH 481
MATH 481: REVIEW 3 (1) A graph is called k-regular if deg(v) = k for any vertex v. What is the size of a k-regular graph of order n? (2) Let G be a graph. The complement G is a graph with the same set of vertices as G, but two vertices are adjacent -
Michigan State University MATH 481
MATH 481: REVIEW 3 (1) A graph is called k-regular if deg(v) = k for any vertex v. What is the size of a k-regular graph of order n? Answer: The size is: 1 2 deg(v) = vV 1 2 k= vV kn 2 (2) Let G be a graph. The complement G is a graph with the s -
Michigan State University MATH 481
MATH 481: HOMEWORK 8 (1) Let G be a graph of order n with vertices v1 , . . . , vn . The line graph L(G) is defined as follows: the vertices of L(G) are the edges of G. If e1 and e2 are two distinct edges of G, we draw an edge between them in L(G) i -
Michigan State University MATH 481
MATH 481: HOMEWORK 9 (1) (a) Find a disconnected graph whose order and size are equal. (b) Find a connected graph that is not complete and not bipartite. (2) Prove that the number of ways to choose k edges from a tree of order n is equal to the numb -
Michigan State University MATH 481
MATH 481: HOMEWORK 9 SOLUTIONS (1) (a) Find a disconnected graph whose order and size are equal. (b) Find a connected graph that is not complete and not bipartite. Answer: (a) K3,3 . (b) The graph of order 4 and size 5, i.e. K4 with an edge removed. -
Michigan State University MATH 481
MATH 481: DISCRETE MATHEMATICS FALL 2008 Course webpage: http:/www.math.msu.edu/dhand/481.html Course Schedule: 8/25 9/1 9/8 9/15 9/22 9/29 10/6 Introduction Holiday No Class 2.2 Binomial Theorem Review 2.4.2 Counting with repetition 2.4.5 Recurrenc -
Michigan State University MATH 481
NOTES FOR MATH 481 THREE BASIC PROBLEMS VIVEK DHAND This material is covered in Section 2.1 of HHM. The purpose of these notes is to reinterpret the three basic problems in terms of maps and to give proofs of the corresponding formulas. Let Bij(X, Y -
Michigan State University MATH 481
NOTES FOR MATH 481 MULTINOMIAL COEFFICIENTS VIVEK DHAND The binomial theorem tells us that (x + y)n = k n n-k k x y k We can rewrite this in a more symmetric way: (x + y)n = a+b=n n! a b x y a!b! where a, b 0. What if we wanted a similar theore -
Michigan State University MATH 481
NOTES FOR MATH 481 THE FIRST THEOREM OF GRAPH THEORY VIVEK DHAND We begin with two proofs of the first theorem of graph theory: the sum of the degrees is equal to twice the number of edges. Theorem. Let G = (V, E) be a graph of order n. Let V = {v1 -
Michigan State University MATH 481
NOTES FOR MATH 481 UNLABELED TREES VIVEK DHAND 1. Labeled and unlabeled trees Recall that a labeled tree of order n is given by a Prfer sequence (a1 , . . . , an-2 ), where u 1 ai n for all i. This shows that there are exactly nn-2 labeled trees o -
Michigan State University MATH 481
MATH 481 REVIEW PROBLEMS 1. 2.1 # 1. 2. (First part of 2.1 # 6) Let be the difference operator (f (x) = f (x + 1) - f (x). Show that (xn ) = nxn-1 Look at pg. 87 for a definition of xn . 3. Prove that {1, 2, . . . , n + 1} has twice as many subsets -
Michigan State University MATH 481
MATH 481: HOMEWORK 3 (1) 2.4.1 # 1. (2) 2.4.1 # 2. (3) 2.4.2 # 1. (4) 2.4.2 # 2. (5) 2.4.2 # 5. -
Michigan State University MATH 481
MATH 481: HOMEWORK 4 (1) 2.4.3 # 1. (2) 2.4.4 # 3. (3) 2.4.4 # 5. a. (4) 2.4.4 # 5. b. (5) 2.4.5 # 3. -
Michigan State University MATH 481
MATH 481: HOMEWORK 4 SOLUTIONS (1) 2.4.3 # 1. Use (25) to compute a1999 , the number of ways to make $19.99 in change. Answer: Since 1999 = 5(399) + 4 we see that a1999 = b399 . Now b399 = 20k+j=399 cj k+5 5 where j 81 because the degree of C(x) -
Michigan State University MATH 481
MATH 481 FALL 2008, MIDTERM EXAM 2 SOLUTIONS 1. How many six-card hands can be dealt from a triple deck? Answer: We need the coefficient of x6 in (1 + x + x2 + x3 )52 = 0ljk 52 k k j j j+k+l x l The possibilities for (l, j, k) such that j + k + -
Michigan State University MATH 481
MATH 481: HOMEWORK 6 In the following problems, let G = (V, E) denote a graph of order n. (1) What is the maximum number of edges of G? (2) If n 2, prove that there exist u, v V such that deg(u) = deg(v). (3) If |E| < n - 1, prove that G is not co -
Michigan State University MATH 481
MATH 481: HOMEWORK 7 (1) Let G be a graph of order n. Find formulas for the order and size of L(G) in terms of the degrees of the vertices of G (see 1.1.3 # 5). (2) Use the previous problem to find the order and size of L(K5 ), and then draw the gra -
Michigan State University MATH 481
MATH 481 REVIEW PROBLEMS FOR EXAM 3 (1) Let G be a k-regular graph of order n. What are the sizes of G and G? (2) Draw the labeled tree whose Prfer sequence is 2,2,2,2. u (3) Prove that L(K1,n ) = Kn . (4) Let G be a graph such that deg(v) 2 for al -
Michigan State University MATH 481
MATH 481: HOMEWORK 10 (1) Find an n n matrix A such that 1 A(i, j) n for all i, j, and there are no repeated entries in any row or column. Use this to find the chromatic number of L(Kn,n ). (2) For each of the following polynomials, explain why i -
Michigan State University MATH 481
MATH 481: HOMEWORK 10 (1) Find an n n matrix A such that 1 A(i, j) n for all i, j, and there are no repeated entries in any row or column. Use this to find the chromatic number of L(Kn,n ). Answer: Here\'s one such 1 2 . . . matrix A (there a -
Michigan State University MATH 481
MATH 481 REVIEW PROBLEMS 1. How many possible outcomes are there if we roll a die fifty times? How about if we roll fifty dice at the same time? Answer: If we roll a die 50 times, we have 50 spots and 6 choices for each spot, so there are 650 possib -
Michigan State University MATH 481
Math 481 Spring 2004 Proof of Burnside\'s Lemma We consider the vertices of a triangle colored with k = 2 colors (R and G). The symmetry group is G = D3 . The table depicts S, the set of all pairs (, c) such that (c) = c. G = (123) 2 = (132) = (1
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