Eulerian graph

An Eulerian graph has a closed trail that traverses every edge exactly once.

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.

A finite graph with at least three vertices is Eulerian exactly when it is connected and every vertex has even degree. A maximal trail closes by parity; closed trails based at vertices with unused incident edges can then be spliced together.