Geometric Structures: Take Home Problems

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/

Problem Set

1. Verify that Desargues Theorem holds for the two triangles:

U = (2, 1, 5); V = (3, 1, 4); W = (1,1, 2)

and

U' = (-1, 0, -2); V' = (-1, 1, 0); W' = (1, 0, 1)

2. Prove or disprove:

G is connected and even-valent but has an edge e such that if e is removed from the graph (e.g. the edge e is erased leaving the vertices at its ends present) the result is a graph which is not connected.

3. Determine all the possible different kinds of plane quadrilaterals that are self-intersecting (e.g. a pair of sides meet at a single point) based on the different partitions for the lengths of the sides and the sizes of angles of the polygon.

4. The graph below illustrates a 3x4 grid graph, where all of the edges are assumed to have the same weight.

Find a formula (with a proof that it works) involving n and m for the minimum number of edges need to eulerize an n x m grid graph. The minimum number of edges to eulerize a graph is the minimum number of edges that must be added to the graph, duplicating existing edges, so that the resulting graph has an Eulerian circuit.