Linear arboricity of robust expanders

TL;DR

This paper proves a longstanding conjecture about decomposing certain dense graphs into linear forests, confirming it for robust expanders with linear minimum degree and related dense graph classes.

Contribution

It confirms the Akiyama-Exoo-Harary conjecture for robust expanders with linear minimum degree, extending to dense quasirandom graphs and large graphs with high minimum degree.

Findings

Conjecture holds for robust expanders of linear minimum degree

Applies to dense quasirandom graphs with high minimum degree

Valid for large graphs with minimum degree close to half the number of vertices

Abstract

In 1980, Akiyama, Exoo, and Harary conjectured that any graph $G$ can be decomposed into at most $\lceil(\Delta(G)+1)/2\rceil$ linear forests. We confirm the conjecture for robust expanders of linear minimum degree. As a consequence, the conjecture holds for dense quasirandom graphs of linear minimum degree as well as for large $n$-vertex graphs with minimum degree arbitrarily close to $n/2$ from above.