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.
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