Eulerian path
In infinite graphs

In an infinite graph, the corresponding concept to an Eulerian trail or Eulerian cycle is an Eulerian line, a doubly-infinite trail that covers all of the edges of the graph. It is not sufficient for the existence of such a trail that the graph be connected and that all vertex degrees be even; for instance, the infinite Cayley graph shown, with all vertex degrees equal to four, has no Eulerian line. The infinite graphs that contain Eulerian lines were characterized by Erdõs, Grünwald & Weiszfeld (1936). For an infinite graph or multigraph G to have an Eulerian line, it is necessary and sufficient that all of the following conditions be met:
change_case:english_lowercase