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ISLaND-hOppING IN KaLININGraD
heard of the seven bridges of königsberg (now known as kaliningrad)? in 1736, the swiss mathematician 
leonhard euler came across a puzzle dealing with these, which many of the inhabitants had tried to solve for 
quite some time. the question was, could you start anywhere in town, cross all seven bridges once, and get back 
where you started? (you can find the layout of the bridges in figure 
5-3
.) to solve the puzzle, euler decided to 
abstract away the particulars and .
 
.
 
. invented graph theory. seems like a good place to start, no?
as you may notice, the structure of the banks and islands in figure 
5-3
 is that of a multigraph; for example, there 
are two edges between a and B and between a and C. that doesn’t really affect the problem. (We could easily 
invent some imaginary islands in the middle of some of these edges to get an ordinary graph.)
What euler ended up proving is that it’s possible to visit every edge of a (multi)graph exactly once and end up 
where you started if and only if the graph is connected and each node has an even degree. the resulting closed 
walk (roughly, a path where you can visit nodes more than once) is called an Euler tour, or Euler circuit, and such 
graphs are Eulerian. (you can easily see that the königsberg isn’t eulerian; all its vertices are of odd degree.)