Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online.In graph theory, an Eulerian path is a path in a graph which visits each edge exactly once. Similarly, an Eulerian circuit is an Eulerian path which starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. Mathematically the problem can be stated like this:Given the graph on the right, is it possible to construct a path (or a cycle, i.e. a path starting and ending on the same vertex) which visits each edge exactly once?Graphs which allow the construction of so called Eulerian circuits are called Eulerian graphs. Euler observed that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and that for an Eulerian path either all, or all but two vertices have an even degree; this means the Königsberg graph is not Eulerian. Sometimes a graph that has an Eulerian path, but not an Eulerian circuit is called semi- Eulerian.Carl Hierholzer published the first complete characterization of Eulerian graphs in 1873, by proving that in fact the Eulerian graphs are exactly the graphs which are connected and where every vertex has an even degree.