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

Abstract

A list assignment of a graph G is a function L that assigns a list L(v) of colors to each vertex $$v\in V(G)$$vźV(G). An $$(L,d)^*$$(L,d)ź-coloring is a mapping $$\pi $$ź that assigns a color $$\pi (v)\in L(v)$$ź(v)źL(v) to each vertex $$v\in V(G)$$vźV(G) so that at most d neighbors of v receive color $$\pi (v)$$ź(v). A graph G is said to be $$(k,d)^*$$(k,d)ź-choosable if it admits an $$(L,d)^*$$(L,d)ź-coloring for every list assignment L with $$|L(v)|\ge k$$|L(v)|źk for all $$v\in V(G)$$vźV(G). In this paper, we prove that every planar graph with neither adjacent triangles nor 6-cycles is $$(3,1)^*$$(3,1)ź-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 $$(3,1)^*$$(3,1)ź-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 $$(3,1)^*$$(3,1)ź-choosable.