1. The accompanying graph has

2. The number of vertices in the following graph is ____, while the number of edges in this graph is ______.

3. The accompanying graph

4. The graph shown below is not connected because it consists of _______ parts.

5. What is the valence of vertex in the graph below?

6. A graph has 24 edges and all the vertices of the graph have valence 3. The number of vertices of the graph must be ______.

7. The valences of the vertices in the accompanying graph, listed in decreasing order, are

8. The alphabetically ordered list of even-valent vertices of the graph below is ______, ______.

9. Which of the statements about the accompanying graph is false?

10. The accompanying graph has _____ edges and ______ vertices.

11. Which of the following statements about a path is true?

12. A graph G has 10 edges, and all its vertices have the same valence. The possible valences of the vertices of G are _____, _____, _____, _____, _____.

13. It is not possible for a graph to have three vertices of valence 3 and six vertices of valence 4 because

14. If a graph consists of five vertices and every pair of vertices is connected by a single edge, the number of edges in the graph is exactly _______.

15. For which of the situations below is it most desirable to find an Euler circuit or an efficient eulerization of the graph?

16. For the accompanying graph, the vertex that has the largest valence is _______, and the number of edges in the graph is _______.

17. The following graph has no Euler circuit because it

18. If a graph is connected and has nine vertices, the graph must have at least _______ edges.

19. Consider the path represented by the sequence of numbered edges on the graph below. Which statement is correct?

20. The accompanying graph has no Euler circuit because it

21. The minimum number of edges that must be duplicated to create a best possible eulerization of the following graph is ______.

22. For the accompanying graph, the minimum total number of edges which constitutes a tour of the graph, starting and ending at the same vertex, and which visits each edge at least once, is _______.

23. 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?

24. If a graph has eight vertices of odd valence, the absolute minimum number of edges that must be added (duplicated) to eulerize the graph is ______.

25. Suppose the edges of a graph represent streets that must be salted after a snowstorm. To eulerize the graph, four edges must be added. The real-world interpretation of this is that

26. For each of the following situations, decide whether a graph or a digraph seems a more reasonable model.

27. The smallest number of edges needed to eulerize the graph below is

28. If the valences of the vertices of a graph G are: 5, 4, 4, 4, 4, 3, 2, 2, and 2, the number of vertices of G is _______ and the number of edges of G is _______.

29. The smallest number of edges that must be added to the accompanying graph for it to have an Euler circuit is

30. The number of edges in a Chinese postman tour (i.e., a tour with a minimum number of edges that starts and ends at the same vertex and visits each edge at least once) for the accompanying graph is ________.