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An Eulerian graph has an Euler circuit: a closed trail traversing every edge exactly once. Here the graph is a finite undirected graph, possibly a multigraph with self-loops. Parallel edges retain distinct identities, and each self-loop contributes two to its vertex degree. Repeated vertices are allowed. This models inspecting every road segment once, even when a junction must be revisited.

For a nonempty set of edges, the Euler circuit criterion requires a connected graph only on the edge support of a graph: isolated vertices impose no travel requirement. A single vertex with a self-loop and two vertices joined by two parallel edges are both Eulerian graphs. Neither requires three vertices. With no edges and at least one vertex, we allow the zero-length trail at that vertex as an Euler circuit; the graph with no vertices has no such walk.

The motivating Seven Bridges of Königsberg problem asks for an Eulerian trail, which need not return to its starting vertex. In Leonhard Euler's Solutio problematis ad geometriam situs pertinentis by Leonhard Euler, §2 poses the bridge-crossing question and §§4–6 encode land regions by letters. The river's shape drops out: only the connections matter. The paper was written in 1735 and published in 1741, according to the Euler Archive; the journal volume carries the nominal year 1736.^[ref]

An Eulerian trail, also called an Eulerian path, is a trail in a graph traversing every edge exactly once. The name Eulerian path does not require a graph path: vertices may repeat. A Hamiltonian path instead visits every vertex exactly once, and may omit edges. An Eulerian trail may omit isolated vertices, so these are different coverage requirements.

For a finite undirected multigraph with at least one edge, allowing self-loops, an Eulerian trail exists exactly when the edge support of a graph is a connected graph and there are zero or two vertices of odd vertex degree. In the two-odd-vertex case they must be the two distinct endpoints; in the zero-odd-vertex case every Eulerian trail is an Euler circuit. Count a self-loop twice in the vertex degree but once in the trail, and count each parallel edge separately.^[ref]

The parity obstruction is local: each intermediate visit pairs an arriving edge with a departing edge. Only the distinct endpoints can have an unpaired incidence. The handshaking lemma, ${\sum }_{v\in V}\mathrm{deg}(v)=2|E|$, also shows that the number of vertices with odd vertex degree is even. Sufficiency in the two-endpoint case reduces to the Euler circuit criterion: add one extra edge between the odd vertices, construct an Euler circuit, rotate its record to start with the extra edge, then delete that edge. The remaining trail has the required endpoints.

For the Seven Bridges of Königsberg multigraph, the vertex degrees at $A,B,C,D$ are $(5,3,3,3)$, so no Eulerian trail exists. Remove the unique edge $AD$: the vertex degrees become $(4,3,3,2)$, and a six-edge Eulerian trail runs from $B$ to $C$. Label the parallel edges $p_1,p_2$ between $A,B$ and $q_1,q_2$ between $A,C$; label $BD$ by $r$ and $CD$ by $s$. A complete record is $B,p_1,A,q_1,C,s,D,r,B,p_2,A,q_2,C$. It revisits $A,B,C$ while using every remaining edge once.