COMPSCI 575/MATH 513: Combinatorics and Graph Theory

Practice Final Exam, Fall 2016

David Mix Barrington

17 December 2016

Directions:

• Answer the problems on the exam pages.
• There are six problems (most with multiple parts) for 125 total points. Estimated scale is A = 110, 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. No explanation is needed or wanted, and there is no partial credit or penalty for guessing (2 points each).
  • (a) Every graph that has a Hamilton circuit also has a Hamilton path that is not a circuit.

  • (b) Every graph that has an Euler circuit also has an Euler path that is not a circuit.

  • (c) Every connected planar graph with 10 vertices has at most 24 edges.

  • (d) The chromatic polynomial of the graph H of Question 5 is r⁴ - 3r³.

  • (e) If every vertex in a tree has odd degree, the number of edges in the tree may be either odd or even.

  • (f) There are exactly ten ways to distribute eleven identical treats to three identical dogs, assuming that each dog gets at least one treat.

  • (g) There are exactly 44 ways permutations of a five-element set that fix none of the elements, that is, 44 derangements of the set.

  • (h) For any non-negative integer n, the sum for i from 0 to n of C(n, i)(-1)ⁱ is equal to 0.

  • (i) There is a game with a finite number of positions whose value, in Conway's terminology, is the number 1/3.

  • (j) Every position in the game of Domineering has a value that is a number, in Conway's terminology.

• Question 2 (20): Recall that an (a, b) cut of a graph G is a partition of the vertices with vertex a in one set and vertex b in the other. In this problem you may quote standard results about network flow without proof.
  • (a, 10) Prove that for any positive integer k, there are k edge-disjoint paths from a to b if and only if every a-b cut contains at least k edges crossing the cut (with one endpoint in each set).

  • (b, 10) Prove that for any positive integer k, there are k vertex-disjoint paths from a to b if and only if every set of vertices, whose removal disconnects a from b, has at least k vertices.

• Question 3 (20): Once again we are distributing identical treats to distinct dogs. Let f(r) be the number of ways to distribute r treats to Cardie, Duncan, and Whistle, with the conditions that (1) each dog gets at least one treat, (2) Cardie gets strictly more treats than Duncan, and (3) Whistle gets no more than three treats.
  • (a, 10) Find f(11) by any valid method (but explain your reasoning).

  • (b, 10) Find the ordinary generating function for the function f, and express this function as a ratio of polynomials (with no "..." or infinite sums). How would you use this generating function to answer part (a)?

• Question 4 (20): For any non-negative integer r, let g(r) be the number of binary strings of length r that do not contain 000 or 111 as a substring, i.e., that never have the same bit three times in a row.
  • (a, 10) Explain why, for r ≥ 3, the function g satisfies the recurrence g(n) = g(n-1) + g(n-2). Give sufficient initial condititions for this recurrence and use it to calculate g(6).

  • (b, 10) Represent an arbitrary binary string w as w₁w₂...w_r. For any positive integer i with i ≤ r-2, let A_i be the set of strings of length r such that w_i = w_i+1 = w_i+2. Use Inclusion/Exclusion on these sets to calculate g(6), and verify that this matches your answer from part (a).

• Question 5 (30): Consider the graph H with vertices {a, b, c, d} and edge set {(a, b), (a, c), (a, d)}.
  • (a, 10) An automorphism of a graph is a graph isomorphism (a bijection h of the vertices such that (u, v) is an edge if and only if (h(u), h(v)) is an edge) from a graph to itself. Prove that H has exactly six isomorphisms.

  • (b, 10) Determine the number of 2-colorings of the vertices of H, where two colorings are considered equivalent if one can be taken to the other by an automorphism of H. Give the pattern inventory for these colorings. Stray text here removed 20 December 2016.

  • (c, 10) Repeat part (b) for 3-colorings of H under the same conditions. You need not simplify the pattern inventory.

• Question 6 (15): Here is a combinatorial game defined on rectangular arrays of stones. Given an n by k array of stones, the player whose move it is may delete either one row of stones, leaving a (k-1) by n array, or delete one column of stones, leaving a k by (n-1) array. As usual, the player taking the last stone wins.
  • (a, 5) Determine the winner, under optimal play, if we start with a 2 by 2 array. You may use any valid method, but justify your answer.

  • (b, 10) Find the kernel of this game, or equivalently, find the exact set of start positions for which the second player has a winning strategy.

Last modified 20 December 2016