For every positive integer k, every finite connected graph G, and every subset S⊆ V(G) such that |delta_G(X)|≥ 2k for every X⊆ V(G) with S cap X≠varnothing and S setminus X≠varnothing, there exist k pairwise edge-disjoint S-Steiner trees T_1,…,T_k in G, where each T_i is a tree satisfying S⊆ V(T_i).

Kriesell’s conjecture

Kriesell’s finite conjecture says that if every edge cut separating a terminal set $T$ has size at least $2k$ , then there are $k$ pairwise edge-disjoint $T$ -Steiner trees. The finite conjecture remains unresolved.

Known results

Kriesell proved the conjecture when every vertex outside $T$ has even degree.
Frank, Király, and Kriesell proved it when the vertices outside $T$ form an independent set.
Lau showed that $24k$ -edge-connectivity suffices.
West and Wu, then Devos, McDonald, and Pivotto, improved the sufficient bound to $6.5k$ and then $5k+4$ .

September 2026 infinite-graph development

Aurichi, Magalhães Júnior, and Monteiro report that an infinite extension is false, while proving positive versions for specified infinite-graph classes and conditional transfers to rayless graphs. This is claimed progress on infinite analogues, not a solution of the finite conjecture, and the retrieved detailed arXiv findings do not independently substantiate these claims.

Current status (as of September 2026): The finite Kriesell conjecture remains open; progress on infinite versions is reported but unverified.

Kriesell’s conjecture

Primary source

Additional references

Progress summary

Solutions 0