A note on (3, 1)-choosable toroidal graphs - Xu, Baogang; Yu, Qinglin (Utilitas Mathematica, 2008)
An (L, d)*-coloring is a mapping Φ that assigns a color Φ(v) ∈ L(v) to each vertex v ∈ V(G) such that at most d neighbors of v receive colore Φ(v). A graph is called (m,d)*-choosable, if G admits an (L,d)*-coloring for every list assignment L with Ł(v) ≥ m for all v ∈ V(G). In this note, it is proved that every toroidal graph, which contains no adjacent triangles and contains no 6-cycles and l-cycles for some l ∈ 5,7, is (3, 1)*-choosable.