COMPSCI 575/MATH 513: Combinatorics and Graph Theory

Final Exam, Fall 2016

David Mix Barrington

22 December 2016

Directions:

• Answer the problems on the exam pages.
• There are six problems (most with multiple parts) for 125 total points. Actual scale was A = 105, C = 70.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• Numerical answers may be given as expressions involving arithmetic operations (including exponentiation and factorial) or the functions P(n, k) and C(n, k). In general a summation is a complete answer to a counting problem, but a recurrence or a generating function is not.

  Q1: 20 points
  Q2: 20 points
  Q3: 20 points
  Q4: 20 points
  Q5: 30 points
  Q6: 15 points
Total: 125 points

• Question 1 (20): Identify each of these statements as true or false. The first five deal with the graph H defined in Question 5. No explanation is needed or wanted, and there is no partial credit or penalty for guessing (2 points each).
  • (a) The chromatic polynomial of H is r⁶ - 7r⁵, plus some terms of dgree less than 5.

  • (b) The graph H has an Euler path but no Euler circuit.

  • (c) The graph H has exactly C(7, 2) = 21 different spanning trees.

  • (d) Consider a weighted undirected graph made from the graph H by giving each edge a distinct weight from the set {1,2,3,4,5,6,7}. It is possible to do this so that the resulting graph has two distinct TSP tours of equal weight. (Recall that a TSP tour is a Hamilton circuit.

  • (e) It is possible to add five, but not six, edges to the graph H (without changing the vertex set) and keep it a planar graph.

  • (f) The exponential generating function for the number of ways to distribute r distinct treats to three distinct dogs, with at least two treats to each dog, is (e^x - x - 1)³.

  • (g) Any function of the form b_r = A2^r, for r ≥ 0, is a solution to the recurrence b_r = b_r-1 + b_r-2 + 2b_r-3, for r ≥ 3.

  • (h) Let G be a two-element group acting on some set S. Then the only variables occurring in the cycle index polynomial of this action are x₁ and x₂.

  • (i) Let G be a game, all of whose left and right options are numbers (in Conway's terminology. Then if every left option is greater than every right option, then G is also a number.

  • (j) In Conway's terminology, let G be a game with finitely many options, all of whose left options are negative and all of whose right options are positive. Then G is a zero game.

• Question 2 (20): Let G be a graph (undirected) with positive integer weights, and let x be a vertex in G. Assume that for every vertex y in G, there is a unique shortest path (defined by using the weights) from x to y. (That is, there are never two different paths with the same minimum weight.) Note that if you claim that a particular tree is an MST for a particular graph, you must justify that claim, either directly or by quoting known results about algorithms.
  • (a, 10) Prove that the union of these paths, over all vertices y in G, is a spanning tree of G.

  • (b, 5) Give an example of a G where this tree fails to be a minimum-weight spanning tree. You may quote known results about algorithsm for the MST problem. Your example should meet the condition stated above about unique shortest paths.

  • (c, 5) Give an example of a G (which could be the same graph as in part (b)) where there is a unique MST T and nodes y and z where the unique path from y to z in T is not the shortest path from y to z in G.

• Question 3 (20): Here is yet another problem where we distribute identical treats to distinct dogs. The rule now is that Cardie gets more treats than Duncan, Duncan gets more than Whistle, and Whistle gets at least one.
  • (a, 5) Determine, by any method, the number of ways to distribute eleven treats to the three dogs while satisfying these rules.

  • (b, 10) Find an ordinary generating function whose x^r coefficient is the number of ways to distribute r identical treats to the three distinct dogs, while following the requirements stated above. Express your generating function as a ratio of polynomials, without any summations or "...". You need not evaluate any coefficients.

    (c, 5) Now consider the problem where the new rule is that each of the three dogs gets a positive integer number of treats, and no two dogs get the same number. Give the ordinary generating function whose x^r coefficient is the number of ways to distribute r identical treats in this way. Express it as a ratio of polynomials.

• Question 4 (20): For any positive integer r, let X_r be the set of strings of length r, over the alphabet {1, 2, 3}, that have all their letters the same. Let Y_r be the set of strings of length r that use exactly two of the three letters. Let Z_r be the set of strings of length r that use all three letters. Let x_r, yr, and z_r be the sizes of X_r, Y_r, and Z_r respectively.
  • (a, 5) Argue that these three functions satisfy the recurrences x_r = x_r-1, y_r = 2x_r-1 + 2y_r-1, and z_r = 3z_r-1 + y_r-1, for all r with r ≥ 2.

  • (b, 5) Solve these recurrences, with appropriate initial conditions, to get formulas for each of these three functions in terms of r.

  • (c, 10) Use Inclusion/Exclusion to derive a formula for z_r, using the three sets A₁, A₂, and A₃, where A_i is the set of strings of length r that have no occurrences of the letter i. Verify that your answer equals your formula found for z_r found in part (b).

• Question 5 (30): Consider the (undirected) graph H with vertex set {A, B, C, D, E, F} and seven edges: (A, B), (A, D), (A, F), (B, C), (C, D), (D, E), and (E, F). Recall that an automorphism is a graph isomorphism from a graph to itself. Also note that the work "coloring" throughout this problem refers to any assignment of colors to the nodes, not just those that are "legal graph colorings". This is the meaning of "coloring" in Chapter 9 of Tucker.
  • (a, 10) Argue that H has exactly four automorphisms. Describe them, and prove that there are no others.

  • (b, 5) Determine the cycle index polynomial of the action of these four automorphisms on the vertices of H.

  • (c, 5) Using part (b) of otherwise, determine the number of 2-colorings of the vertices of H, where two colorings are considered the same if any automorphism takes one to the other.

  • (d, 5) Determine the pattern inventory for 2-colorings of H as in part (c), where the vertices are colored black (b) or white (w). You may leave your answer as a polynomial expression without placing it in simplest form.

  • (e, 5) Describe the equivalence classes of 2-colorings of H that have four white and two black vertices. The number of these should match your answer to part (d).

• Question 6 (15): In this problem we have a two-player combinatorial game played with four stones arranged in a 2 by square array. On each turn, the player to move chooses a row or a column of the array that has at least one stone, and removes at least one stone from that row or column. (If there are two stones, she may remove either or both.) As usual, the winner is the player who removes the last stone.
  • (a, 5) Who has a winning strategy for this game? Prove your answer (which may follow from your answer to parts (b) and (c)).

  • (b, 5) We know from Chapter 9 of Tucker that up to symmetry, there are exactly six possible positions in this game. List these positions and determine, for each one, who wins the game starting from that position. (Thus you should determine the kernel of this game.)

  • (c, 5) Determine the Grundy number of each of the six positions. That is, for each position p, determine the number of stones in a single pile whose Nim game is equivalent to this game starting from p.

Last modified 10 January 2017