= Euler circuit
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= Euler tour
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An <Euler circuit> is a closed edge trail that traverses every edge exactly once. Vertices may repeat, and parallel edges are counted separately. A connected finite undirected multigraph has such a circuit exactly when every vertex has even degree. Necessity follows by pairing arrivals and departures; sufficiency follows by forming closed unused-edge trails and splicing them until every edge is used, as in the <Euler circuit criterion>.
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