MTH 105 – Review for Test 1
1.
Define the following terms and draw an example for each.
a. complete graph
b. bipartite graph
c. simple graph
d. disconnected graph
e. regular graph
f. connected graph
2. Determine the number of edges in a complete graph with 10 vertices.
3. Determine the number of vertices in a
complete graph with 66 edges.
4. Complete the following sentences:
a. A graph has an Euler circuit if and only if _________________________________________________.
b. A graph has an Euler path if and only if __________________________________________________.
5. What is the difference between an Euler circuit and a Hamilton circuit?
6. Add the least number of edges to create an Euler circuit in the graph (Eulerization, optimal route). Label the edges to indicate the Euler circuit.
7.
For each graph below, determine if it has the indicated properties.
a.
b.
Euler circuit? ____
Euler path? ____
Hamilton circuit? ____
Hamilton path? ____
c.
Euler circuit? ____
Euler path? ____
Hamilton circuit? ____
Hamilton path? ____
d.
Euler circuit? ____
Euler path? ____
Hamilton circuit? ____
Hamilton path? ____
Euler circuit? ____
Euler path? ____
Hamilton circuit? ____
Hamilton path? ____
8. Are the graphs in 7a and 7d isomorphic? Why or why not?
9. Determine which pairs of graphs are isomorphic.
a.
b.
c.
d.
e.
f.
10. Given the following weighted graph, find the following tours:
A
B
6
a. Nearest­Neighbor starting at A
11
3
10
______________________________________
8
2
b. Nearest­Neighbor starting at B
7
E
9
C
4
5
______________________________________
c. Cheapest­Link
______________________________________
D
11. Given the following map, draw its dual graph, G, and determine (G).
12. Explain why  = 2 for all bipartite graphs.