Eulerian graph

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One area of graph theory is concerned with the possibility of travelling around a graph, going along edges in such a way as to use every edge exactly once. A connected graph is called Eulerian if there is a sequence v0, e1, v1,…, ek, vk of alternately vertices and edges (where ei is an edge joining vi−1 and vi), with v0=vk and with every edge of the graph occurring exactly once. Simply put, it means that ‘you can draw the graph without taking your pencil off the paper or retracing any lines, ending at your starting-point’. The name arises from Euler's consideration of the problem of whether the bridges of Königsberg could be crossed in this way. It can be shown that a connected graph is Eulerian if and only if every vertex has even degree.

Subjects: Mathematics.

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