Graph Coloring: Definition, Algorithm, and Application in Exam Scheduling, Slides of Data Structures and Algorithms

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Graph Coloring

Definition

• A coloring of a simple

graph is the

assignment of a color

to each vertex of the

graph so that no two

adjacent vertices are

assigned the same

color.

What is the chromatic number of this graph?

Algorithm

• Assign color 1 to the vertex with highest degree.
• Also assign color 1 to any vertex that is not connected to this vertex.
• Assign color 2 to the vertex with the next highest degree that is not already colored.
• Also assign color 2 to any vertex not connected to this vertex and that is not already colored.
• If uncolored vertices remain, assign color 3 to the uncolored vertex with next highest degree and other uncolored, unconnected vertices.
• Proceed in this manner until all vertices are colored.

Application

• e.g. Scheduling Final Exams
• Suppose you want to schedule final exams and,

being very considerate, you want to avoid having

a student do more than one exam a day. We shall

call the courses 1,2,3,4,5,6,7. In the table below a

star in entry ij means that course i and j have at

least one student in common so you can't have

them on the same day. What is the least number

of days you need to schedule all the exams? Show

how you would schedule the exams.

Day Exam 1 1, 5 2 2, 4 3 3, 6 4 7