List vertex-arboricity of toroidal graphs without 44-cycles adjacent to 33-cycles

The vertex-arboricity a(G)a(G) of a graph GG is the minimum number of colors required to color the vertices of GG such that no cycle is monochromatic. The list vertex-arboricity al(G)al(G) is the list-coloring version of this concept. Kronk and Mitchem (1975) proved that every toroidal graph GG without 33-cycles has a(G)≤2a(G)≤2. Choi and Zhang (2014) proved that every toroidal graph GG without 44-cycles has a(G)≤2a(G)≤2. Borodin and Ivanova (2009) proved that every planar graph GG without 44-cycles adjacent to 33-cycles has al(G)≤2al(G)≤2. In this paper, we improve and extend these results by showing that al(G)≤2al(G)≤2 if GG is a toroidal graph without adjacent 3- and 4-cycles.