# Orientation conjecture of Nash-Williams

From Egres Open

Any [math] 2k [/math]-edge-connected (possibly infinite) multigraph admits a [math] k [/math]-edge-connected orientation.

## Remarks

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 ^{[1]}. The conjecture is implied by its own restriction to countably infinite, locally finite multigraphs hence it is essentially a countable problem.

## References

- ↑ C. Thomassen,
*Orientations of infinite graphs with prescribed edge-connectivity*, Combinatorica (2014): 1-21. DOI link, author link