**Combinatorial and Discrete Geometry: Sheet Z (Vertex
Colorings)**

Prepared by:

Joseph Malkevitch

Department of Mathematics and Computer Studies

York College
(CUNY)

Jamaica, New York 11451

email: joeyc@cunyvm.cuny.edu

web page: http://www.york.cuny.edu/~malk

Vertex colorings
arise in a variety of natural problems including the problem of
scheduling final examinations for a high school or college so that no
exams are for the same time for courses which have common students.
Thus, we will not have a situation where a student must be in two different places
at the same time. Can you find other examples of situations where
vertex colorings avoid "conflicts?"

For each of the
graphs below determine if possible a coloring with 4 colors. Also
determine the chromatic number for the graphs.