Final Exercises

Final Exercises - 1 Math 2602 M1/M2 Spring 2007 Additional...

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Unformatted text preview: 1 Math 2602 M1/M2 Spring 2007 Additional Problems for Final Exam Comprehensive 1 Thursday, April 25th 1. A plotter is a device for drawing pictures such as one in the figure below. A plotter starts drawing by placing its pen on the paper and moving it until the entire picture has been drawn. At that moment, the pen is raised and the drawing of the picture is finished. After that, the paper is shifted and another copy of the same picture is drawn. Because of this, it is demanded that the point where the drawing starts and the point where it ends are the same. Note that the pen of the plotter can move through some lines of the figures several times. The time necessary to draw a figure is proportional to the total distance of moves of the pen. In order to maximize the efficiency of the device, this distance should be the least possible. v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v 12 v 13 v 14 v 15 v 16 v 17 Your task is to formulate the problem of finding an optimal way of drawing the picture as one of the graph theory problems introduced during the lecture. You should also explain why the problem models the situation. Give an optimum solution of the problem you have just described. You are not required to show that the solution you have found is an optimal one. 2. Determine the chromatic numbers of the following three graphs. In all the cases, you are required to construct a coloring with the least number of colors and argue that there is no coloring with fewer colors. 1 T. Yolcu, School of Mathematics at Georgia Tech. Email: tyolcu@math.gatech.edu . 2 3. An electricity company wants to interconnect five town in Graph county: town A, town B, town C, town D and town E. It is not necessary to build a direct link between all the pairs of the towns, but it will be enough to be possible to transport the electric power between every pair of towns using links leading through other towns in the county....
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Final Exercises - 1 Math 2602 M1/M2 Spring 2007 Additional...

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