2⁵ Problems for STEM Education by Ochkov Valery;

Author:Ochkov, Valery;
Language: eng
Format: epub
Publisher: CRC Press LLC

The graph of the Königsberg bridges had four odd vertices (see Figure 15.12), therefore, it was impossible to pass through all bridges, without passing through either of them twice.

FIGURE 15.12 Simplified diagram and graph of Königsberg bridges*.

* http://eulerarchive.maa.org//docs/originals/E053.pdf.

The theory of the graphs created by Euler (thanks to this problem) has found very wide application: for example, it is used in studying transport and communication systems and, in particular, for routing data on the Internet. Note that the traveling salesman’s problem also admits a solution because its graph (see Figures 15.9 and 15.11) has all the vertices even (the entrance to the city along one road, the exit along the other, that is, exactly two edges). This graph obeys the second assertion of Euler. However, the presence of a solution does not mean at all that this solution will be perceived as natural and intuitive. For example, the traveling salesman’s problem contains a paradox: often he should not go to the nearest city!

And here there are three more problems, similar to the traveling salesman’s problem and also having elements of paradox.

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