New bounds for linear arboricity and related problems

TL;DR

This paper improves bounds on decomposing graphs into linear forests, showing that fewer than previously thought linear forests suffice, using novel rotation techniques, with implications for spanning forests and graph tours.

Contribution

It introduces a new method generalising Pósa rotations to achieve tighter bounds on linear arboricity, resolving longstanding conjectures.

Findings

Proves $rac{ ext{Δ}}{2} + ext{O}( ext{log } n)$ linear forests suffice for decomposition.

Provides exponential improvement over previous bounds when $ ext{Δ} = ext{Ω}(n^ ext{ε})$.

Resolves a conjecture on spanning linear forests with few paths and short tours in regular graphs.

Abstract

A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta+1)/2\rceil$ linear forests. We prove that $\Delta/2 + \mathcal{O}(\log n)$ linear forests suffice, where $n$ is the number of vertices of the graph. If $\Delta = \Omega(n^\varepsilon)$, this is an exponential improvement over the previous best error term. We achieve this by generalising P\'osa rotations from rotations of one endpoint of a path to simultaneous rotations of multiple endpoints of a linear forest. This method has further applications, including the resolution of a conjecture of Feige and Fuchs on spanning linear forests with few paths and the existence of optimally short tours in connected regular graphs.