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.