A note on (3, 1)-choosable toroidal graphs

Abstract

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.