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1 Graph Theory Name: Study Guide Mods: Date: Define each of the following. It may be helpful to draw examples that illustrate the vocab word and/or counterexamples to define the word. 1. Graph ~ 2. Vertex ~ 3. Edge ~ 4. Connected Graph ~ 5. Non-connected Graph ~ 6. Path ~ 7. Circuit ~ 8. Euler Path ~ 9. Euler Circuit ~ 10. Valence ~ 11. complete graph ~ 12. Eulerize a Graph ~ 13. Hamiltonian Circuit ~ 14. algorithm ~ 15. Brute Force Method ~ 16. method of trees ~ 17. fundamental counting principle ~ 18. Nearest Neighbor Algorithm ~ 19. Sorted Edges Algorithm ~ 20. Traveling Salesman Problem ~ 21. Chromatic Number ~ 22. Four-color map problem ~

2 CHAPTER 1: 1. What is the valence of vertex A in the graph below? 2. The valences of the vertices in the accompanying graph listed in decreasing order are (a) 3 (b) 2 (c) 4 (a) 1, 1, 1, 2, 3, 3, 4, 5 (b) 5, 4, 4, 2, 1, 1, 1. (c) 5, 4, 3, 3, 2, 1, 1, The graph below is not connected because 4. A vertex in the graph below that is even-valent is (a) it has all even-valent vertices. (b) it consists of two parts. (a) C. (b) A. (c) E. (c) it consists of three parts. 5. The graph below has 6. Which of the following statements is true about a path? (a) A path always forms a circuit. (a) four vertices and six edges. (b) four vertices and four edges. (b) A path is always connected. (c) A path can visit any vertex only once. (c) five vertices and six edges. 7. If a graph consists of four vertices and every pair of vertices is connected by a single edge, how many edges are in the graph? 8. It is not possible for a graph to have five vertices of valence 3 and six vertices of valence 4 because (a) there are no graphs with exactly 11 vertices. (b) a graph cannot have an even number of 4-valent (a) four (b) five (c) six vertices. (c) a graph cannot have an odd number of odd-valent vertices.

3 9. If a graph is connected and has six vertices, what can be said about the number of edges in the graph? 10. Consider the path represented by the sequence of numbered edges on the graph below. Which statement is correct? (a) There are at least five edges in the graph. (b) There are exactly five edges in the graph. (c) There are at least six edges in the graph. (a) The sequence of numbered edges forms an Euler circuit. (b) The sequence of numbered edges traverses each edge exactly once but is not an Euler circuit. (c) The sequence of numbered edges forms a circuit. 11. For the graph below, which statement is correct? 12. Suppose each vertex of a graph represents a baseball team and each edge represents a game played by two baseball teams. If the resulting graph is not connected, which of the following statements must be (a) The graph has an Euler circuit. (b) One new edge is required to eulerize the graph. (c) Three new edges are required to eulerize the graph. true? (a) At least one team never played a game. (b) At least one team played every other team. (c) The teams play in distinct leagues. 13. Suppose the edges of a graph represent streets that must be plowed after a snowstorm. To eulerize the graph, four edges must be added. The real-world interpretation of this is that (a) four streets will not be plowed. (b) four streets will be traversed twice. (c) four new streets would be built. 14. If the vertices of a graph represent cities and the edges of a graph represent flight routes for a particular airline, then which of the objects below best models a pilot s daily schedule? (a) A path (b) An Euler circuit (c) A graph 15. In the graph below, add one or more edges to produce a graph that has an Euler circuit.

4 CHAPTER 2 1. Which of the following describes a Hamiltonian circuit for the graph below? 2. Using the nearest-neighbor algorithm and starting at vertex A, find the cost of the Hamiltonian circuit for the graph below. (a) ABCDEA (b) ABEDCBDAC (c) ACDEBA 3. The nearest-neighbor traveling salesman tour for the following graph starting at B is (a) 25 (b) 27 (c) Using the sorted-edges algorithm, find the cost of the Hamiltonian circuit for the graph below. (a) BCDAB (b) BDCAB (c) BCADB 5. Suppose that after a hurricane, a van is dispatched (a) 25 (b) 26 (c) Another answer 6. The graph below has to pick up five nurses at their homes and bring them back to work at the local hospital. Which of these techniques is most likely to be useful in solving this problem? (a) Finding an Euler circuit in a graph (b) Solving a TSP (traveling salesman problem) (c) Finding a minimum-cost spanning tree in a graph 7. Apply the nearest-neighbor method (starting at (a) no Hamiltonian circuit and no Euler circuit. (b) an Euler circuit and a Hamiltonian circuit. (c) no Hamiltonian circuit, but it has an Euler circuit. 8. Apply the sorted-edges method to find a cheap tour. vertex A) to find a cheap tour.

5 CHAPTER 3 1. Given the order-requirement digraph below (time in minutes) and the priority list T 1, T 2, T 3, T 4, T 5, T 6, apply 2. A vertex coloring seeks to color the vertices of a graph to ensure which of the following traits? the list-processing algorithm to construct a schedule using two processors. How much time does the resulting schedule require? (a) Every color is used. (b) Every edge connects vertices of the same color. (c) Vertices of the same color are never connected by (a) 11 minutes (b) 13 minutes (c) 14 minutes 3. The minimum number of colors needed to color the vertices of the accompanying graph is an edge. 4. Determine the minimum number of colors, and how often each color is used, in a vertex coloring of the graphs below: (a) 4 (b) 2 (c) 3 T 1, T 2, T 3, T 4, T 5, T 6

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