Fractional coloring problem of 1-planar graphs without short cycles

We consider a new coloring as follows such that a subset ofV(G) needs very few colors.LetGbe a graph with vertex setV(G)anda, b, ibe three integers witha >= b >= i >= 0.Sis a vertex subset ofG,graphGis said to bei-weak (a:b)-choosable about setS,ifthefollowing can be satisfied: For any list assignmentL={L(v):|L(v)|=a, v is an element of V(G)},there is a way that each vertexvofSis assignedicolors ofL(v) and each vertexuofV(G)\Sis assignedbcolors ofL(v) such that the color sets corresponding to adjacentvertices do not intersect. In this paper, we consider thei-weak (a:b)-choosable problemof 1-planar graphs without short cycles and prove our results by the discharging method.In this paper, we prove that every 1-planar graph without 3-cycles, 5-cycles and adjacent4-cycles isi-weak (4b+i:b)-choosable about setS={v:v is an element of V(G),d(v)=4}