A [[Graph Trail|trail]] is eulerian iff it is a closed trail (circuit) that includes all edges one time. A [[Graphs|graph]] is eulerian if it contains an eulerian trial. A trail is semi-eulerian iff it is an open trail that includes all edges one time. A graph is semi-eulerian if it is not eulerian and contains a semi-eulerian eulerian trial. # Properties A graph is eulerian iff all vertices has even degree. Proof. A graph is eulerian iff exactly two vertices have odd degree. Proof.