Independent spanning trees conjecture

Prove or disprove the conjecture that every k-vertex-connected graph contains k independent spanning trees rooted at any specified vertex for all k≥6.

Background

Independent spanning trees would provide a potential replacement for the tree-packing tools used in the edge-fault-tolerant setting. The cited conjecture is known for connectivity parameters up to 5, but remains unresolved for all larger parameters; even if proved, it would directly apply only under a strong vertex-connectivity assumption.

References

It was conjectured by that every $k$-vertex connected graph contains a collection of $k$ independent spanning trees for every root $r$. They proved this conjecture for $k=2$, and it has subsequently been proved for $k=3$, $k=4$, and $k=5$, but it is still open for all $k \geq 6$.

— Light Edge Fault Tolerant Graph Spanners  (2502.10890 - Bodwin et al., 15 Feb 2025) in Section 5, paragraph “Independent Spanning Trees”

It was conjectured by that every $k$-vertex connected graph contains a collection of $k$ independent spanning trees for every root $r$. They proved this conjecture for $k=2$, and it has subsequently been proved for $k=3$, $k=4$, and $k=5$, but it is still open for all $k \geq 6$.

— Light Edge Fault Tolerant Graph Spanners  (2502.10890 - Bodwin et al., 15 Feb 2025) in Section 5, paragraph “Independent Spanning Trees”