The Linear Arboricity of Graphs on Surfaces of Negative Euler Characteristic

Abstract

The linear arboricity of a graph $G$ is the minimum number of linear forests which partition the edges of $G$. In the present, it is proved that if a graph $G$ can be embedded in a surface of Euler characteristic $\varepsilon<0$ and $\Delta(G)\geq\sqrt{46-54\varepsilon}+19$, then its linear arboricity is $\lceil\frac{\Delta(G)}{2}\rceil$. Some related results on the girth and maximum average degree are also obtained.