A note on the list vertex arboricity of toroidal graphs

The vertex arboricity $a\left(G\right)$ of a graph $G$ is the minimum number of colors required to color the vertices of $G$ such that no cycle is monochromatic. The list vertex arboricity $a_l\left(G\right)$ is the list-coloring version of this concept. In this note, we prove that if $G$ is a toroidal graph, then $a_l\left(G\right)\le 4$; and $a_l\left(G\right)=4$ if and only if $G$ contains $K_7$ as an induced subgraph.