Mathematical_Recreations-Kraitchik-2e

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Mathematical Recreations
bridges is given in Figure 98. If we shrink each piece of land
to a point and each bridge to a line, we transform the geographic
map into what is called a linear graph, shown in
Figure 99. This particular graph has four odd verticesvertices
at which an odd number of lines come together. It
can be shown that it is impossible to pass just once over every
arc of a linear graph without jumping if the graph has more
than two odd vertices. Hence it is impossible to fulfill the
prescription of the Konigsberg bridges problem.
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FIGURE 98. Geographic Map: FIGURE 99.
The KOnigsberg Bridges.
Linear Graph.
The problem is sometimes stated in the following form:
A smuggler wishes to cross just once every frontier of each
of the various countries of a certain continent. What is his
route? If we represent the various countries by points and
join by simple arcs every pair of points representing countries
with a common frontier, the problem is seen to be of the
same type. If the continent is that of Europe no solution is
possible, since several countries, Denmark and Portugal for
example, have an odd number of frontiers.
Every linear graph has an even number of odd vertices.
If there are no odd vertices the graph is unicursal and
B