[mhandley]

"Euler noticed that if all the nodes in the graph have an even number of edges (such graphs are called “Eulerian” in his honor) then, and only then, a cycle can be found that visits every edge exactly once."

Unfortunately the article is not quite correct. All non-terminal nodes must have an even number of edges, but the first and last nodes in the cycle can have an odd number.

[detaro]

How does a cycle have a first and last node?

[mhandley]

Yes, you're correct - I shouldn't have said "cycle". I was refering to an Eulerian path, which is all you need to solve the Bridges of Konigsberg problem as it is described in the article. A cycle adds the constraint that the path starts and finishes at the same place, and then all the nodes must have even degree. But you don't need a cycle to just cross all the bridges once.