Magnant–Martin path-cover conjecture

Prove that the vertices of every d-regular graph of order n can be covered by at most n/(d + 1) vertex-disjoint paths.

Background

This conjecture seeks an optimal upper bound on the number of vertex-disjoint paths required to cover all vertices of a regular graph. It is connected to the linear arboricity conjecture and to path-partition results for regular graphs.

The paper explicitly notes that the conjecture remains open. Its later application proves only an asymptotic version for sufficiently large degree, allowing uncovered vertices and using cycles instead of paths.

References

This latter conjecture is still open; see for some interesting recent progress towards it.

— Nearly Hamilton cycles in sublinear expanders, and applications  (2503.07147 - Letzter et al., 10 Mar 2025) in Section 1, subsection “Packing subgraphs in regular graphs”