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 pi 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?