**Final Examination Problem Set**

Prepared by:

Joseph Malkevitch

Department of Mathematics

York College (CUNY)

Jamaica, New York 11451

email:

__malkevitch@york.cuny.edu__

web page:

__http://york.cuny.edu/~malk/__

Enjoy!

1. Show that the Petersen Graph contains the graph K_{3,3} as a subdivision, that is, can be obtained from a copy of K_{3,3} by introducing 2-valent vertices along edges of that graph.

2. a. Show that there can be no 3-valent 3-polytopal graph whose faces are all 13-gons and 3-gons.

b. Can there be a 3-valent 3-connected 3-polytopal graph with exactly two 13-gons and the remaining faces 3-gons and 6-gons? Explain your answer.

3. Show that there exists a value of p_{6}* such that there exists a plane 3-valent 3-polytopal graph G with p_{3}(G) = 6, p_{12}(G) = 1 and p_{6}(G) = p_{6}*.

4. Construct a planar 4-valent 3-connected graph for which: p_{3} = 35 and p_{13} = 3, and any other faces present are 4-gons.

5. Show that it is possible to "shave" an edge joining two 3-valent vertices of a plane 3-polytopal graph G which has no hamiltonian circuit and obtain a graph G* which does have a hamiltonian circuit.

The diagram below illustrates an edge shave to edge e:

After the edge shave to graph G (which lacks a hamiltonian circuit) to get graph G* (which has a hamiltonian circuit) for the graph you use, can a hamiltonian circuit of G* use only the opposite sides of the 4-gon on the right?

6. Assume each edge of the graph below has weight 1. What is the cost of a tour (closed walk) which starts at vertex 4, traverses each edge of this graph at least once and returns to vertex 4, and whose total cost is as small as possible? Find a tour of the graph that achieves this minimum.

7. Prove or disprove that given positive integers that solve:

but for which t_{2}=0, that there is a planar 3-connected plane graph all of whose faces are triangles which has a tree with these valences as a spanning tree.

7. Show for each integer n greater than or equal to 3 that K_{n} is NOT the visibility graph of some simple plane non-convex polygon with n sides.

(Two vertices of a plane simple polygon P are visible if the line joining them contains no points of the exterior of the polygon.)

8. State and prove rigorously the graph theory version of Euler's Polyhedral Formula for plane graphs.

9. Determine the vertex, face, and edge coloring numbers of the graph:

10. a. Verify that the graph below is the union of cut-through circuits, and, thus, is not cut-through Eulerian.

b. Verify that the graph below has an Eulerian circuit which is not cut-through at any vertex.

c. Verify that there is a 4-valent 3-polytopal graph which has two 6-gons and twelve triangles, which is cut-through Eulerian for some choice of 4-gons for the graph.

11. Construct a new problem or conjecture which you believe has not yet been answered, based on what you have learned in this course.

*Extra*

1. Show that there exists a value of p_{6}* such that there exists a plane 3-valent 3-polytopal graph G with p_{3}(G) = 11, p_{13}(G) = 3 and p_{6}(G) = p_{6}*.