Review For Final Examination (Spring 2005)

**Mathematics 244 (Geometric Structures)**

1. Draw a diagram of the following (when possible the diagram should be drawn in the plane):

a. A scalene right triangle

b. An equiangular triangle

c. An equiangular 4-gon that is not a square

d. An equilateral quadrilateral that is not a square

e. A non-convex, non-self-intersecting 4-gon

f. A trapezoid with three equal sides that is not a rectangle

g. An equiangular hexagon which is not a regular hexagon

h. A self-intersecting regular pentagon

i. An isosceles right triangle

j. An acute angle triangle

k. A parallelogram which is not a rectangle or a rhombus

l. A rhombus which is not a square

m. A quadrilateral with exactly two, nonadjacent right angles

n. Draw a graph with 10 vertices and 14 edges

o. Draw an equiangular triangle

p. Draw an isosceles triangle which is not equilateral

q. A non-convex self-intersecting 7 sided polygon

r. Give some examples of 3-dimensional geometrical figures

s. Give some examples of 2-dimensional geometrical figures

t. What is the difference between a ray and a segment?

u. Draw graph which is planar but not plane.

v. Draw a graph which is non-planar.

2. Give a precise statement of the graph theory version of Euler's formula.

3. Draw a graph to which one can not apply Euler's formula.

4. a. Write the p_{i} values for the faces of the graph below.

b. Is there a pair of vertices u and v in the graph above such that there are three paths between these vertices?

c. Is the graph above 3-connected?

5. Draw an example of a 3-valent (e.g. every vertex is 3-valent) which is 3-polytopal.

6. Draw an example of a 4-valent graph (e.g. every vertex is 4-valent) which is 3-polytopal.

7. Draw a plane graph with every vertex having valence at least 3 which is not 3-polytopal.

8. Give a statement of the four color conjecture.

9. Can one color the faces of the graph in Problem 4 above with exactly 3 colors?