Feige–Fuchs linear-order path-cover conjecture

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

Background

The Feige–Fuchs conjecture is a weaker asymptotic form of the Magnant–Martin conjecture: it permits an unspecified absolute constant factor in the number of paths.

Despite being weaker and following from the linear arboricity conjecture for all d, the paper states that it remains widely unresolved.

References

Even the much weaker conjecture by Feige and Fuchs that every $d$-regular graph of order $n$ can be covered by at most $O(n/(d + 1))$ vertex-disjoint paths (which follows from the linear arboricity conjecture for all $d$) remains wide open.

— 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”