Orientation conjecture of Nash-Williams
Any [math] 2k [/math]-edge-connected (possibly infinite) multigraph admits a [math] k [/math]-edge-connected orientation.
The conjecture is formulated by Nash-Williams who also proved the restriction of it to finite multigraphs; see Nash-Williams' strong orientation theorem. The first (and so far the only) breakthrough is due to C. Thomassen. He showed that a [math] 8k [/math]-edge-connected infinite multigraph admits a [math] k [/math]-edge-connected orientation . The conjecture is implied by its own restriction to countably infinite, locally finite multigraphs hence it is essentially a countable problem.