### Chapter Test

For the following Exercises, use the figure shown.

1.

Name the edges in Graph

*T*.2.

Identify the graph(s) with six edges.

For the following exercises, use the figure shown.

3.

Identify any of the graphs that is a subgraph of Graph

*D*.4.

Identify any graphs with no cyclic subgraphs of any size.

5.

Consider Graph 4 and Graph 6 in the given figure. Determine if one graph is a subgraph of the other. If so, give a correspondence between vertices that demonstrates this relationship. If not, identify conflicting characteristics.

6.

For the following exercise, use Graph

*A*in the given figure. Consider each sequence of vertices. Determine if it is only a walk, both a walk and a path, both a walk and a trail, all three, or none of these.*d*â†’*e*â†’*b*â†’*c*â†’*f*7.

For the following exercise, use Graphs

*A*and*K*in Figure 12.369. Determine the chromatic number of Graph*K*. Give a coloring that supports your conclusion.For the following exercise, use the graphs and multigraphs in the given figure.

8.

Identify any graphs and/or multigraphs that are not Eulerian. If there are none, state so.

9.

List the set of vertices for each component in Graph 11.

10.

Find an Euler circuit beginning and ending at vertex

*b*in Graph 12 if one exists.11.

Use Fleuryâ€™s algorithm to construct an Euler trail for the given graph beginning at vertex

*f*of your choice.12.

Use the graphs shown to determine whether the sequence of vertices

*d*â†’*b*â†’*c*â†’*f*â†’*e*â†’*d*is a Hamilton cycle, an Euler circuit, both, or neither.13.

Calculate the number of distinct Hamilton cycles in a complete graph with 15 vertices.

14.

Use the figure shown to find the weight of the given Hamilton cycle:

*t*â†’*w*â†’*x*â†’*u*â†’*y*â†’*v*â†’*s*â†’*r*â†’*q*â†’*t*For the following exercises, use this information: For the Halloween celebration at an elementary school, the students from Classroom

*I*will visit every classroom once and return to their own classroom. The floor plan of the school is given.15.

Draw a graph to represent the classrooms of the elementary school in which each vertex is a classroom and an edge between two vertices indicates that there is a path between the two rooms that does not pass the door to another classroom.

16.

Use your graph to find a Hamilton circuit beginning and ending at

*I*.17.

Explain what the Hamilton circuit you found represents for the students in Classroom

*I*.18.

Find a Hamilton cycle of low weight for Graph 18, beginning at vertex

*q*, and using the nearest neighborhood method. What is the weight of the cycle?19.

Use Kruskalâ€™s Algorithm to find a minimum spanning tree for the below graph. Graph the tree and give its weight.