(3, 1)-Choosability of toroidal graphs with some forbidden short cycles

Lih et al. showed that every planar graph without 4-cycles or ii-cycles for some i∈{5,6,7}i∈{5,6,7} is (3,1)(3,1)-choosable. Dong and Xu showed that every toroidal graph without 4-cycles or 6-cycles is (3,1)(3,1)-choosable. In this paper, we show that every toroidal graph without 4-cycles or ii-cycles for some i∈{5,7}i∈{5,7} is also (3,1)(3,1)-choosable.