Orientation conjecture of Nash-Williams

Any $$2k$$ -edge-connected (possibly infinite) multigraph admits a $$k$$ -edge-connected orientation.

Remarks

The conjecture is formulated by Nash-Williams who also proved the restriction of it to finite multigraphs; this known as Nash-Williams' weak orientation theorem. See Nash-Williams' strong orientation theorem for the strong version of the theorem. The first (and so far the only) breakthrough for infinite graphs is due to C. Thomassen. He showed that a $$8k$$ -edge-connected infinite multigraph admits a $$k$$ -edge-connected orientation . The conjecture is implied by its own restriction to countably infinite, locally finite multigraphs hence it is essentially a countable problem.