1. A busy airport has 1500 takeo s perday. Provetherearetwoplanesthatmusttake o within one minute of each other. This is from Bona Chapter 1 (1).

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1 Math/CS 415 Combinatorics and Graph Theory Fall 2017 Prof. Readdy Homework Chapter 1 1. A busy airport has 1500 takeo s perday. Provetherearetwoplanesthatmusttake o within one minute of each other. This is from Bona Chapter 1 (1). 2. Assume 169 points are selected at random inside of a regular triangle of side length 100. Prove that among these points three span a triangle of area at most A set M concists of nine positive integers with the property that none have a prime divisor larger than 6. Prove M has two elements whose product is the square of an integer. 4. Prove that among any 502 positive integers there are always two integers so that either their sum or di erence is divisible by (Extra credit) Let a i be the integer consisting of i 7 s. Determine a value of i so that the integer a i is divisible by the prime Chapter 2 6. Prove integers of the form a(n) =n 3 +11n for n 1aredivisibleby6. 7. Prove by induction that 2 n(n +1) n 3 = Cut a square into four smaller squares, then cut some of the obtained small squares each into four smaller squares. Repeat the process. Prove that at any given point of time during this operation, the number of squares is of the form 3m +1wherem is some nonnegative integer. 9. Define a sequence a n by a 0 =3anda n = a 0 a 1 a n 1 +2 for n 1. Prove a n =2 2n Define the Fibonacci numbers by F 0 =1,F 1 =1andF n = F n 1 + F n 2 for n 2. Make a conjecture about when F n is divisible by 7 and then prove it.

2 Chapter How many functions are there from {1,...,n} to {1,...,n} that are not 1-1? 12. Baskin Robbins advertises 31 flavors of ice cream. How many di erent two scoop cones can they make, assuming order does not matter? What if order of the scoops matters? 13. How many three-digit positive integers contain two but not three di erent digits? 14. How many ways are there to list the letters of the word ALABAMA? 15. a. Determine the number of subsets of {1,...,n} containing the elements 1 and 2. b. Determine the number of subsets of {1,...,n} containing at least one of the elements 1and2. Chapter The Powerball lottery consists of selecting five numbers from the integers 1 through 69 and one extra integer from 1 through 26. How many possible powerball tickets can one buy? 17. Prove 2n n is even. 18. Find a closed form for f(x) = P 1 n=1 nxn Give two di erent proofs of 3 n = nx n 2 k k k=0 20. Prove that if a sequence of positive numbers {a i } i 0 = a 0,a 1,a 2,...is log-concave then it is unimodal. Chapter How many compositions does the integer 15 have where the first part is not Find the number of weak compositions of 25 into five odd parts. 23. Find the number of set partitions of {1,...,8} into two blocks. 24. a. Find the number of partitions of the integer n into odd parts for n =1,...,7. b. Find the number of partitions of the integer n into distinct parts for n =1,...,7. c. Do you notice anything about parts a and b? 25. Devise a card trick using the Bell number B(4).

3 Chapter For the following permutations written in one-line notation, express each as the product of disjoint cycles. a. w = b. z = Express as the product of disjoint cycles: a. (1, 2, 3)(4, 5)(1, 6, 7, 8, 9)(1, 5) b. (1, 2)(1, 2, 3)(1, 2) 28. Prove that the inverse of (1, 2,...,n)is(n, n 1,n 2,...,2, 1). 29. Find the cycle structure of all the powers of g =(1, 2, 3, 4, 5, 6, 7, 8). Chapter Use the Principle of Inclusion and Exclusion to determine the number of positive integers less than or equal to 210 which are relatively prime to 210. (Hint: factor 210.) 31. Use PIE to count the number of permutations in the symmetric group S 8 having descent set {1, 4, 6}. 32. Use PIE to count the number of compositions of n where neither the first or last part equals 1. Chapter Let b n be the number of compositions of n where each part is an odd integer. a. Find a closed formula for the ordinary generating function P n 0 b nx n. b. Explain why the generating function in part a shows b n = F n F n 2 = F n 1,the n 1st Fibonacci number. 34. Let a n be the number of ways to pay n dollars using $10 bills, $5 bills and $1 bills. a. Find the ordinary generating function A(x) = P n 0 a nx n. b. Use your generating funciton to find the coe cient of x 25. You may use a computer to find this. Chapter Show how the city of Köningsberg could build two additional bridges and then have a closed Eulerian walk. 36. Find two non-isomorphic simple graphs with the same degree sequence. 37. Determine which of the following pictures can be traced without lifting your pencil. (See enclosed page.)

4 Graph Tracing Puzzles Trace these graphs by starting at any vertex and drawing each edge once without lift your pencil from the paper. Can you use the trick to know where to start? Which one impossible? Why?

5 38. Let G be the graph on the vertices V = {a, b, c, d, e} and edges E = {(a, b), (a, d), (a, e), (b, c), (c, d), (d, e)}. a. Draw the graph G. b. Is G planar? c. Give the adjacency matrix of G. d. How many paths are there of length 5 from the vertex b to the vertex e? Chapter Given a graph G its complement G satisfies V (G) =V (G) and{u, v} is an edge in G if and only if {u, v} is not an edge in G. a. Prove that if G is a simple graph on n vertices then at least one of G and its complement is connected. b. Find an example of a graph where it and its complement are both connected. 40. Draw the labeled rooted tree whose Prüfer code is (1, 1, 1, 6, 1, 5). 41. Find the Prüfer code of the labeled rooted tree below: 42. a. Find the number of spanning trees in the complete bipartite graphs K m,n. b. Let A be the adjacency matrix of K m,n. Explain which entries of A k must be zero for any positive integer k. 43. Find a spanning tree of maximal weight for the graph in Figure 10.5 of your text (page 222).

6 Chapter a. Show K 3,3 is a planar graph when drawn on the torus. b. Show K 7 is a planar graph when drawn on the torus. 45. A Platonic solid is a polytope where all of its faces are congruent convex regular polygons and the same number of faces meet at each vertex, that is, f 0,2 is some fixed constant. One often describes Platonic solids using its Schläfli symbol {p, q}, where each facet is a p-gon and q facets are incident with each vertex. a. Verify that p f 2 =2 f 1 = q f 0. b. Use Euler s relation to determine the possible values of the Schläfli symbol {p, q} for a 3-dimensional Platonic solid. (There are five. Can you name them?) 46. Prove that the coe cients of the chromatic polynomial of a graph alternate in sign. 47. Calculate the chromatic polynomials of the following two graphs:

7 Chapter 10 again Use the Four Color Theorem to conclude that the graph K 5 is not planar. 49. Use the Matrix Tree Theorem to count the number of spanning graphs of the graph in problem 47 on the left-hand side if we were to label the vertices. (See previous page.) 50. Use the Matrix Tree Theorem to show the number of spanning trees for the graph K n e, where e is any edge of the complete graph, is (n 2) n n 3.