A $$(3,1)^{*}$$-choosable theorem on planar graphs

A list assignment of a graph G is a function L that assigns a list L(v) of colors to each vertex (vin V(G)). An ((L,d)^*)-coloring is a mapping (pi ) that assigns a color (pi (v)in L(v)) to each vertex (vin V(G)) so that at most d neighbors of v receive color (pi (v)). A graph G is said to be ((k,d)^*)-choosable if it admits an ((L,d)^*)-coloring for every list assignment L with (|L(v)|ge k) for all (vin V(G)). In this paper, we prove that every planar graph with neither adjacent triangles nor 6-cycles is ((3,1)^*)-choosable. This is a partial answer to a question of Xu and Zhang (Discret Appl Math 155:74–78, 2007) that every planar graph without adjacent triangles is ((3,1)^*)-choosable. Also, this improves a result in Lih et al. (Appl Math Lett 14:269–273, 2001) which says that every planar graph without 4- and 6-cycles is ((3,1)^*)-choosable.