A note on the list vertex arboricity of toroidal graphs

The vertex arboricity a(G) 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 al(G) is the list-coloring version of this concept. In this note, we prove that if G is a toroidal graph, then al(G)≤4; and al(G)=4 if and only if G contains K7 as an induced subgraph.