Linear arboricity conjecture

Prove that every graph with maximum degree Δ can be decomposed into at most ⌈(Δ + 1)/2⌉ path forests.

Background

The linear arboricity conjecture asks for the minimum number of path forests needed to decompose the edges of a graph in terms of its maximum degree. The conjectured bound is the natural analogue of classical arboricity-type decomposition results.

The paper presents this as a prominent unresolved problem related to vertex coverings by disjoint paths, but does not address or resolve it.

References

Perhaps the most famous open problem in this area is the linear arboricity conjecture of Akiyama, Exoo, and Harary from 1980, which states that every graph with maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta + 1)/2\rceil$ path forests.

— 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”

Perhaps the most famous open problem in this area is the linear arboricity conjecture of Akiyama, Exoo, and Harary from 1980, which states that every graph with maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta + 1)/2\rceil$ path forests.

— 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”

Perhaps the most famous open problem in this area is the linear arboricity conjecture of Akiyama, Exoo, and Harary from 1980, which states that every graph with maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta + 1)/2\rceil$ path forests.

— 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”