List vertex-arboricity of toroidal graphs without 4-cycles adjacent to 3-cycles

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 a l ( G ) is the list-coloring version of this concept. Kronk and Mitchem (1975) proved that every toroidal graph G without 3 -cycles has a ( G ) ź 2 . Choi and Zhang (2014) proved that every toroidal graph G without 4 -cycles has a ( G ) ź 2 . Borodin and Ivanova (2009) proved that every planar graph G without 4 -cycles adjacent to 3 -cycles has a l ( G ) ź 2 . In this paper, we improve and extend these results by showing that a l ( G ) ź 2 if G is a toroidal graph without adjacent 3- and 4-cycles.