2 7 Se p 20 06 A Note on ( 3 , 1 ) ∗ - Choosable Toroidal Graphs †

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 |L(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.