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 ~

CHAPTER 1:

1. What is the valence of vertex A in the graph below?

(a) 3 (b) 2 (c) 4

2. The valences of the vertices in the accompanying

graph listed in decreasing order are

(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, 1.

3. The graph below is not connected because

(a) it has all even-valent vertices.

(b) it consists of two parts.

(c) it consists of three parts.

4. A vertex in the graph below that is even-valent is

(a) C. (b) A. (c) E.

5. The graph below has

(a) four vertices and six edges.

(b) four vertices and four edges.

(c) five vertices and six edges.

6. Which of the following statements is true about a

path?

(a) A path always forms a circuit.

(b) A path is always connected.

(c) A path can visit any vertex only once.

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?

(a) four (b) five (c) six

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

vertices.

(c) a graph cannot have an odd number of odd-valent

vertices.

9. If a graph is connected and has six vertices, what

can be said about the number of edges in the graph?

(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.

10. Consider the path represented by the sequence of

numbered edges on the graph below. Which statement

is correct?

(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?

(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.

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

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.

CHAPTER 2

1. Which of the following describes a Hamiltonian

circuit for the graph below?

(a) ABCDEA (b) ABEDCBDAC (c) ACDEBA

2. Using the nearest-neighbor algorithm and starting at

vertex A, find the cost of the Hamiltonian circuit for

the graph below.

(a) 25 (b) 27 (c) 26

3. The nearest-neighbor traveling salesman tour for the

following graph starting at B is

(a) BCDAB (b) BDCAB (c) BCADB

4. Using the sorted-edges algorithm, find the cost of

the Hamiltonian circuit for the graph below.

(a) 25 (b) 26 (c) Another answer

5. Suppose that after a hurricane, a van is dispatched

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

6. The graph below has

(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.

7. Apply the nearest-neighbor method (starting at

vertex A) to find a cheap tour.

8. Apply the sorted-edges method to find a cheap tour.

CHAPTER 3 1. Given the order-requirement digraph below (time in

minutes) and the priority list T1, T2, T3, T4, T5, T6, apply

the list-processing algorithm to construct a schedule

using two processors. How much time does the resulting

schedule require?

(a) 11 minutes (b) 13 minutes (c) 14 minutes

2. A vertex coloring seeks to color the vertices of a

graph to ensure which of the following traits?

(a) Every color is used.

(b) Every edge connects vertices of the same color.

(c) Vertices of the same color are never connected by

an edge.

3. The minimum number of colors needed to color the

vertices of the accompanying graph is

(a) 4 (b) 2 (c) 3

4. Determine the minimum number of colors, and how

often each color is used, in a vertex coloring of the

graphs below:

T1, T2, T3, T4, T5, T6