Euler trails and Euler circuits in multigraphs and digraphs

In a finite undirected multigraph, a walk of length $\ell$ is an alternating list

$v_0,e_1,v_1,e_2,\ldots,e_\ell,v_\ell$

such that the endpoint set assigned to $e_i$ is $\{v_{i-1},v_i\}$ for each $i$. Thus the particular edge is recorded, parallel edges remain distinguishable, and $v_{i-1}=v_i$ precisely when that step uses a loop. The walk is closed when $v_0=v_\ell$, and it is a trail when the displayed edges are distinct.

An Euler trail is a trail that uses every edge exactly once. An Euler circuit is a closed Euler trail, and a multigraph is Eulerian when it has an Euler circuit. A length-zero closed trail at the sole vertex of the edgeless one-vertex multigraph is an Euler circuit.

In a finite digraph, a directed Euler trail is a directed trail that uses every arc exactly once. A directed Euler circuit is a closed directed Euler trail, and a digraph is directed Eulerian when it has one. The simple-graph walk conventions are in Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges, and the directed-trail terminology is that of Directed walks, trails, paths and cycles, and strong connectivity.