Mathematics 243 (Combinatorial and Discrete Geometry)

Review: Examination II

prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451

web page:

1. a. Draw a tree whose only vertices are 1-valent and 5-valent, and has 3-valent vertices. How many edges does this tree have?

b. Draw a tree which has vertices of valence 1, 3, 5, and 7 and no vertices of other valences.

c. Some of the properties of a graph that must hold if the graph is a tree but will not hold if the graph is not a tree.

2. For the graphs below use:

a. Kruskal

b. Prim

c. Boruvka (use this method only when the edges all have different weights)

to find a minimum cost spanning tree for the graph as well as the cost associated with that tree:

3. Draw a connected weighted graph G which has cycles and for which the two most expensive edges in G are present in any minimum cost spanning tree for G.

4. Give some examples of real world situations where the finding a minimum cost spanning tree would be required.

5. For each graph below give if possible a proper 3-coloring of the graph, and determine the vertex chromatic number of the graph and give a proper coloring that uses the chromatic number of colors:

6. For each graph in Problem 5 which is a plane graph, wrote down the degree sequence of the graph and the face vector of the graph. (The face vector records the values pi which give the number of faces of the graph which have i sides.)

7. Draw a planar graph with 6 vertices which is bipartite, connected, has a cycle of length 5, and is not a plane graph.

8. For each graph below:

a. If possible find a matching with 5 edges.

b. If possible find a maximal matching with 5 edges.

c. If possible find a perfect matching.

d. Find a maximum cardinality M