New bounds for linear arboricity and related problems

A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $\mathrm{\Delta }$ can be decomposed into at most $\lceil (\mathrm{\Delta }+1)\mathrm{/}2\rceil$ linear forests. We prove that $\mathrm{\Delta }\mathrm{/}2+\mathcal{O}(\log n)$ linear forests suffice, where $n$ is the number of vertices of the graph. If $\mathrm{\Delta }=\mathrm{\Omega }(n^{\epsilon })$, this is an exponential improvement over the previous best error term. We achieve this by generalising Pósa 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.