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

The vertex arboricity ρ(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 ρl(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 ρl(G)≤2 for any toroidal graph without 5-cycles‎. ‎We also show that ρl(G)≤2 if G contains neither adjacent 3-cycles nor cycles of lengths 6 and 7.