# Well-balanced orientations of hypergraphs

When can we characterize hypergraphs that have an orientation satisfying a prescribed symmetric local edge-connectivity requirement? Special case: can we characterize hypergraphs that have an orientation which is *k*-edge-connected within a specified subset of nodes?

## Remarks

Frank, Király and Király^{[1]} proved an extension of Nash-Williams' weak orientation theorem by characterizing hypergraphs which have a *k*-edge-connected orientation. In this result "orienting a hypergraph" means picking a head node from each hyperedge.

For the same definition of hypergraph orientation, Király and Lau^{[2]} proved that if a hypergraph *H* is *2k*-edge-connected within a subset *T* of nodes, then for any [math]r \in T[/math] it has an orientation where from any node of *T* there are *k* edge-disjoint paths to *r*.

To extend the strong orientation theorem of Nash-Williams^{[3]} to hypergraphs, one may need a different definition of hypergraph orientation. This may be related to the open problem on Highly element-connected orientation of graphs.

## References

- ↑ A. Frank, T. Király, Z. Király,
*On the orientation of graphs and hypergraphs*, Discrete Applied Mathematics. 131 (2003) 385--400. DOI link. EGRES Technical Report no. 2001-06 - ↑ T. Király, L.C. Lau,
*Approximate min-max theorems for Steiner rooted-orientations of graphs and hypergraphs*, Journal of Combinatorial Theory B 98 (2008), 1233-1252. DOI link. EGRES Technical Report no. 2006-13 - ↑ C. St. J. A. Nash-Williams,
*On orientations, connectivity and odd vertex pairings in finite graphs*, Canad. J. Math. 12 (1960) 555-567.