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

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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.