(1,0,0)-Colorability of planar graphs without prescribed short cycles

Let be non-negative integers. A graph is -colorable, if the vertex set of can be partitioned into subsets such that the subgraph induced by has maximum degree at most for . Let be the family of planar graphs with cycles of length neither 4 nor 8. In this paper, we prove that a planar graph in is -colorable if it has no cycle of length for some . Together with other known related results, this completes a neat conclusion on the -colorability of planar graphs without prescribed short cycles, more precisely, for every triple , planar graphs without cycles of length 4, or are -colorable whenever 4 < i < j <= 9.