A new sufficient condition for a toroidal graph to be 4-choosable

A graph G is k-choosable if G can be properly colored whenever every vertex has a list of at least k available colors. In this paper, we will proof that if every 5-cycle of toroidal graph G is not adjacent simultaneously to 3-cycles and 4-cycles, then G is 4-choosable. This improves a result shown in Xu and Wu (2017), which stated that if every 5-cycle of planar graph G is not adjacent simultaneously to 3-cycles and 4-cycles, then G is 4-choosable.