The Linear Arboricity Conjecture

Let $G$ be a simple graph, let $la(G)=a_2(G)$ be its linear arboricity, and let $\Delta(G)$ be its maximum degree. A linear forest is a forest whose connected components are paths.

Linear Arboricity Conjecture. For every simple graph $G$,

$$la(G)\le\left\lceil\frac{\Delta(G)+1}{2}\right\rceil.$$

The lower bound $la(G)\ge\lceil\Delta(G)/2\rceil$ makes this an additive-one assertion. The source reports proofs for several graph classes and asymptotic results, but the conjecture is not stated as resolved in general.