The following graph has 9 vertices. a. Does this graph have an Euler circuit? Why? 22 21 12 b. Does this graph have an Euler path? Why? 20 02 10 01 c. Does this graph have a Hamilton path? Why? 00

13. Consider the following graph: B F E Does it have an Euler circuit, an Euler path, or neither? Explain why.

a. the number of edges in the graph, b. the number of vertices in the graph, c. the number of vertices that are of odd degree, d. whether the graph is connected, and e. whether the graph is a complete graph. 5.

Is it possible to have an Euler Circuit in the above diagram? Justify.Is it possible to have an Euler Circuit in the diagram below? If there is an Euler Circuit, what is the path?

13. A mail person decides to park their vehicle and walk through one of the neighborhoods they deliver mail. They want to return to their vehicle without crossing any path twice. Can they get back to their vehicle using the graph K, below? Find the degrees of the vertices, then find an Euler circuit if one exists. h K9 Fill in the degree of each vertex based on the graph. Vertex Degree a b d e f g y Euler circuit:

The graph shown has at least one Euler circuit. Determine an Euler circuit that begins and ends with vertex C. Complete the path so that it is an Euler circuit. C, A, B, E, D, A, 0 ... (.... D B E

B Complete the following for the graph on the right. a. Determine if the graph must have Hamilton circuits. G b. If the graph must have Hamilton circuits, determine the number of such H circuits. K ..... a. Must the graph have Hamilton circuits? O A. Yes, it must because it is complete and has at least three vertices. O B. Yes, it must because it is not complete. O C. No, it might not because it is complete and has at least three vertices. OD. No, it might not because it is not complete. b. How many circuits, if any, does the graph have? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The graph has Hamilton circuits. O B. The graph does not have any Hamilton circuits.

13. The edges in the figure below represent electrical lines that must be inspected and the vertices represent junctions. Is there a route that will not require the inspector to travel along already inspected lines? That is, is there an Euler circuit for the graph in the Figure? Be sure to explain your answer. B E