COMPSCI 575/MATH 513: Combinatorics and Graph Theory

First Midterm Exam, Fall 2016

David Mix Barrington

11 October 2016

Directions:

• Answer the problems on the exam pages.
• There are eight problems (some with multiple parts) for 100 total points. Actual scale was 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, f} and twelve edges: (a, b), (a, d), (a, e), (a, f), (b, c), (b, d), (b, e), (c, d), (c, e), (c, f), (d, f), and (e, f).

• Question 1 (10): Are all graphs with six nodes and twelve edges isomorphic to H? Prove 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 the chromatic number of H? (This is the "vertex-chromatic" rather than the "edge-chromatic" number.) Prove your answer.

• Question 5 (15): Here are two questions about "graph-theoretic" trees, by which we mean connected acyclic (undirected) graphs.
  • (a, 5) Using the average degree of a node, argue that every tree has at least one leaf. (This should have said "every tree with n ≥ 2 nodes".)

  • (b, 10) Prove by induction that for all integers n with n ≥ 2, every tree with n nodes has at least two leaves.

• Question 6 (15): Let J be the following 16-node weighted (undirected) graph. The nodes are pairs (i, j) where 0 ≤ i, j ≤ 3. There is an edge from (i, j) to (i', j') if and only if i = i' and |j=j'| = 1 or |i-i'| = 1 and j = j'. (The nodes thus form a Manhattan-style grid enclosing nine blocks.) The weight of the edge between (i, j) and (i', j'), where i > i' or j *gt; j', is 1 if i+j is odd and 2 if i+j is even.
  • (a, 10) Determine the weight of a minimum spanning tree of J.

  • (b, 5) Generalize this graph to a graph J_n, where the graph J above is J₃, so that J_n has (n+1)² nodes. Find and justify a formula for the weight of an MST for J_n.

• Question 7 (15): Consider a weighted bipartite (undirected) graph where the nodes are divided into two sets A and B, each of size n. We asserted but did not prove in lecture that there is an algorithm with time polynomial in n, that inputs such a graph a returns a matching of minimum possible weight. (A matching is a subset S of the edges such that each node is an endpoint of exactly one edge in S.)

  Suppose we instead wanted a matching with the maximum possible total weight. Indicate how we can use the given algorithm for minimum-weight matching to solve this problem in polynomial time.

• Question 8 (15): Let G be a network with positive edge capacities, a source a, and a sink z. Assume that G has no edge from a to z. For any positive number c, let G_c be the network obtained from G by adding a new edge from a to z with capacity c.
  • (a, 10) State and prove a result relating the maximum possible flow in G to that in G_c.

  • (b, 5) Explain why, given the result of part (a), we may assume in our analysis of maximum flows that there is no edge from a to z.

Last modified 22 October 2016