On list vertex 2-arboricity of toroidal graphs without cycles of specific length

The vertex arboricity $rho(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ can be partitioned so that each subset induces an acyclic graph‎. ‎A graph $G$ is called list vertex $k$-arborable if for any set $L(v)$ of cardinality at least $k$ at each vertex $v$ of $G$‎, ‎one can choose a color for each $v$ from its list $L(v)$ so that the subgraph induced by every color class is a forest‎. ‎The smallest $k$ for a graph to be list vertex $k$-arborable is denoted by $rho_l(G)$‎. ‎Borodin‎, ‎Kostochka and Toft (Discrete Math‎. ‎214 (2000) 101-112) first introduced the list vertex arboricity of $G$‎. ‎In this paper‎, ‎we prove that $rho_l(G)leq 2$ for any toroidal graph without 5-cycles‎. ‎We also show that $rho_l(G)leq 2$ if $G$ contains neither adjacent 3-cycles nor cycles of lengths 6 and 7.