COMPSCI 575/MATH 513: Combinatorics and Graph Theory

Practice First Midterm Exam, Fall 2016

David Mix Barrington

4 October 2016

Directions:

• Answer the problems on the exam pages.
• There are eight problems (some with multiple parts) for 100 total points. Estimated scale is A = 93, C = 63.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 15 points
  Q6: 15 points
  Q7: 15 points
  Q8: 15 points
 Total: 100 points

For Questions 1-4, the graph H has node set {a, b, c, d, e} and nine edges, consisting of all possible edges except (c, e).

• Question 1 (10): Are all graphs with five nodes and nine edges isomorphic to H? Justify your answer.

• Question 2 (10): Is H a planar graph? Prove your answer. What is the number of regions in a planar embedding of H, given by Euler's Formula?

• Question 3 (10): Does H have an Euler path and/or Euler circuit? Does H have a Hamilton path and/or Hamilton circuit? Justify your answers.

• Question 4 (10): What is H's chromatic number? (This is the "vertex-chromatic" rather than the "edge-chromatic" number.) Prove your answer.

• Question 5 (15): Prove that if G is any graph, either G is connected or G's complement G^C is connected. (G^C is a graph with the same vertices as G and exactly the edges that are not in G.) (Hint: Use induction on n, the number of vertices, and consider the possible cases.)

• Question 6 (15): Let K be a weighted (undirected) complete graph with n vertices, where every edge weight is either 1 or 2. Let K' be the ordinary graph with the same vertices as K and an edge wherever the corresponding edge of K has weight 1. Suppose that K' has a Hamilton circuit. What is the weight of a minimum spanning tree of K? Prove your answer (that is, prove both that it is not too high and not too low).

• Question 7 (15): Suppose we had an approximation algorithm for TSP that, on graphs with n nodes, always found a Hamilton circuit whose weight was at most n times the weight of the optimal circuit. Indicate how we could use this algorithm to solve the NP-complete Hamilton circuit problem. (Hint: Your argument should involve weighted graphs that do not obey the triangle inequality.)

• Question 8 (15): Suppose we have a network with one or more sources, one or more sinks, positive integer capacities on the directed edges, and intermediate nodes with zero net flow. The value of a flow in such a network is the sum of the flows out of the sources. Describe how we can use the results about ordinary networks (with one source and one sink) to find the maximum possible flow in this new network,

Last modified 22 October 2016